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* Scientific Library | 










LORD KELVIN, G.C.V.O. D.C.L. LL.D. F.R.S. &c. 



11 Nee aranearum sane textus ideo melior quia ex se fila gignunt, nee noster 
vilior quia ex alienis libaraus ut apes." Just. Lips. Polit. lib. i. cap. 1. Not. 







f*1 O /"* O i*-" 

- .- » ;.- > 


" Meditation is est perscrutari occulta; contemplationis est admirari 
perspicua .... Adniiratio generat quaestionem, quaestio iuvestigationem, 
investigatio inventionem." — Hugo de 8. Victore, 

" Cur spirent venti, cur terra dehiscat, 

Cur mare turgescat, pelago cur tantus amaror, 
Cur caput obscura Phoebus ferrugine condat, 
Quid toties diros cogat flagrare conietas, 
Quid pariat nubes, yeniant cur fuluiina coelo, 
Quo micet igne Iris, superos quis conciat orbes 
Tam vario motu." 

J. B. Tinelli ad Mazonium. 





- Page 

Mr. Albert Campbell on the Magnetic Fluxes in Meters and 
other Electrical Instruments 1 

Messrs. Edward B. Eosa and Arthur W. Smith on a Eeson- 
ance Method of Measuring Energy dissipated in Con- 
densers 19 

Dr. E. H. Cook on Experiments with the Brush Discharge. 
(Plate I.) 40 

Dr. van Eijckevorsel on the Analogy of some- Irregularities 
in the Yearly Range of Meteorological and Magnetic 
Phenomena. (Plate II.) 57 

Lord Kelvin on the Age of the Earth as an Abode fitted for 
Life 66 

Mr. D. L. Chapman on the Eate of Explosion in Gases .... 90 

Prof. Carl Bar us on the Aqueous Fusion of Glass, its Eelation 
to Pressure and Temperature 104 

Prof. E. Eutherford on Uranium Eadiation and the Electrical 
Conduction produced by it 109 

Notices respecting New Books : — 

Dr. E. J. Eouth's Treatise on Dynamics of a Particle . . 163 

Susceptibility of Diamagnetic and Weakly Magnetic Sub- 
stances, by A. P. Wills 164 


Dr. Thomas Preston on Eadiation Phenomena in the Mag- 
netic Field. — Magnetic Perturbations of the Spectral 
Lines 165 



Lord Kelvin on the .Reflexion and Refraction of Solitary Plane 
Waves at a Plane Interface between two Isotropic Elastic 
Medinms— Fluid, Solid, or Ether 179 

Prof. H. L. CalleDdar : Notes on Platinum Thermometry . . 191 

Messrs. Edward B. Rosa and Arthur W. Smith on a Calori- 
metric Determination of Energy dissipated in Condensers . 222 

Prof. Karl Pearson on certain Properties of the Hyper- 
geometrical Series, and on the fitting of such Series to 
Observation Polygons in the Theory ot Chance 236 

Lord Rayleigh on James Bernoulli's Theorem in Probabilities . 246 

Notices respecting New Books : — 

Drs, G. E. Eisher and I. J. Schwatt's Textbook of Algebra 
with Exercises for Secondary Schools and Colleges . . 251 

Relative Motion of the Earth and JEther, by "William 
Sutherland 252 


Prof. J. J. Thomson on the Theory of the Conduction of 

Electricity through Gases by Charged Ions 253 

Mr. William Sutherland on Cathode, Lenard, and Rontgen 

Rays 269 

Dr. R. A. Lehfeldt on Properties of Liquid Mixtures.— 

Part III. Partially Miscible Liquids 284 

Mr. W. B. Morton on the Propagation of Damped Electrical 

Oscillations along Parallel Wires 296 

Lord Kelvin on the Application of Sellmeier's Dynamical 
Theory to the Dark Lines D,, D 2 produced by Sodium 

Vapour " 302 

Lord Rayleigh on the Cooling of Air by Radiation and Con- 
duction, and on the Propagation of Sound 308 

Lord Rayleigh on the Conduction of Heat in a Spherical 

Mass of Air confined by Walls at a Constant Temperature. 314 
Notices respecting New Books : — 

Dr. D. A. Murray's Elementary Course in the Integral 

Calculus ." 325 

Proceedings of the Geological Society : — 

Mr. J. E. Marr on a Conglomerate near Melmerby 

(Cumberland) , 326 

Mr. Beeby Thompson on the Geology of the Great 

Central Railway, Rugby to Catesby . . , 327 

Prof. T. T. Groom on the Geological Structure of the 
Southern Malverns and of the adjacent District to 
the West 327 


On the Heat produced by Moistening Pulverized Bodies : 
New Therm ometrical aDd Calorimetrical Kesearches, by 
Tito Martini 329 

Combinatiou of an Experiment of Ampere with an Experi- 
ment of Faraday, by J. J. Taudin Chabot 331 

Experiments with the Brush Discharge, by E. W. Marchant . 331 


Dr. C. Chree on Longitudinal Vibrations in Solid and 

Hollow Cylinders 333 

Mr. R. "W. Wood on some Experiments on Artificial Mirages 

and Tornadoes. (Plate III.) 349 

Mr. J. Eose-Innes and Prof. Sydney Young on the Thermal 

Properties of Normal Pentane 353 

Mr. R. W. Wood on an Application of the Diffraction- 
Grating to Colour-Photography 368 

Dr. G. Johnstone Stoney on Denudation and Deposition. . . . 372 
Lord Eayleigh on the Transmission of Light through an 
Atmosphere containing Small Particles in Suspension, and 

on the Origin of the Blue of the Sky 375 

Prof. Oliver Lodge on Opacity 385 

Prof. J. J. Thomson : Note on Mr. Sutherland's Paper on the 

Cathode Pays 415 

Notices respecting New Books : — 

Harper's Scientific Memoirs, edited by Dr. J. S. Ames. . 417 
Proceedings of the Geological Society : — 

Mr. W. Wickbam King on the Permian Conglomerates 

of the Lower Severn Basin 417 

Dr. Maria M. Ogilvie [Mrs. Gordon] on the Torsion- 
Structure of the Dolomites 419 


Mr. F. H. Pitcher on the Effects of Temperature and of 
Circular Magnetization on Longitudinally Magnetized Iron 
Wire 42L 

Dr. Edwin H. Barton on the Equivalent Resistance and 
Inductance of a Wire to an Oscillatory Discharge 433 

Mr. L. N. G. Filon on certain Diffraction Fringes as applied 
to Micrometric Observations 441 


Prof. Carl Barus on the Absorption of Water in Hot Glass 

(Second Paper) 461 

Lord Kelvin on the Application of Force within a Limited 
Space, required to produce Spherical Solitary Waves, or 
Trains of Periodic Waves, of both Species, Equivoluminal 

and Irrotational, in an Elastic Solid 480 

Dr. C. Chree on Denudation and Deposition 494 

Notices respecting New Books : — 

Prof. Silas Holman's Matter, Energy, Force, and Work. 496 
Messrs. J. Harkness and F. Morley's Introduction to the 

Theory of Analytic Functions 497 

Proceedings of the Geological Society: — 

Mr. H. H. Arnold-Bemrose on the Geology of the Ash- 
bourne and Buxton Branch of the London and North- 
Western Railway — Ashbourne to Crake Low 498 

Prof. J. B. Harrison and Mr. A. J. Jukes-Browne on 

the Oceanic Deposits of Trinidad 498 

A Five-cell Quadrant Electrometer, by Prof. H. Haga .... 499 


Mr. Edwin S. Johonnott, Jun., on the Thickness of the Black 
Spot in Liquid Films , 501 

Mr. Albert Griffiths on the Source of 'Energy in Diffusive 
Convection 522 

Mr. Albert Griffiths : Study of an Apparatus for the 
Determination of the Rate of Diffusion of Solids dissolved 
in Liquids 530 

Mr. G. A. Shakespear on the Application of an Interference- 
Method to the Investigation of Young's Modulus for Wires, 
and its Relation to Changes of Temperature and Magneti- 
zation; and a further Application of the same Method to 
the Study of the Change in Dimensions of Iron and Steel 
Wires by Magnetization 539 

Dr. G. Johnstone Stoney on Denudation and Deposition. — 
Part II 557 

Mr. Gerald Stoney on the Quantity of Oxygen in the Atmo- 
sphere, compared with that in the Earth's Crust 565 

Lord Rayleigh on the Calculation of the Frequency of Vibra- 
tion of a System in its Gravest Mode, with an example 
from Hydrodynamics 566 

Notices respecting New Books : — 

Dr. F. A. Tarleton's Introduction to the Mathematical 
Theory of Attraction 572 



H. Poin care's Theorie du Potentiel Newtonien 573 

Dr. Elorian Cajori's History of Physics in its Ele- 
mentary Branches, including the Evolution of Physical 

Laboratories 575 

Proceedings of the Geological Society : — 

Mr. Prank Eutley on a small Section of Felsitic Lavas 

and Tuffs near Conway (North. Wales) 575 

Mr. Joseph Thomson on the Geology of Southern 
Morocco and the Atlas Mountains 576 

Index 577 


T. Illustrative of Dr. E. H. Cook's Paper on Experiments with the Brush 


II. Illustrative of Dr. van Pijckevorsel's Paper on the Analogy of some 

Irregularities in the Yearly Range of Meteorological and Mag- 
netic Phenomena. 

III. Illustrative of Mr. R. W. Wood's Paper on some Experiments on 

Artificial Mirages and Tornadoes. 






JANUARY 1899. 

I. The Magnetic Fluxes in Meters and other Electrical 
Instruments, By Albert Campbell, B.A* 

IN all electrical measuring instruments in which the de- 
flecting or controlling forces are electromagnetic, the 
magnetic fluxes and fields are of great importance, and yet 
there seem to be no tables published which give even rough 
measurements of these, the result being that many people 
who are thoroughly expert in the use of instruments have 
no idea whatever of the order of magnitude of the magnetic 
fluxes occurring in the very commonest instruments. In 
order, therefore, to fill this gap to some extent, I have 
recently carried out a series of experiments on the subject, 
and although the list of instruments thus tested is not very 
extensive or complete, I have been able to include in it a 
good many of the more familiar types. As individual instru- 
ments of the same type vary somewhat amongst themselves, 
it would have been waste of time to have aimed at great 
accuracy in these measurements. Accordingly, whilst guard- 
ing against large errors in general, I have been content in 
one or two cases with results which only indicate the order 
of magnitude of the quantity measured. 

In most cases the quantity determined has been B, the 
magnetic flux density or number of induction-tubes per square 
centimetre sometimes through iron, sometimes through air. 

* Communicated by the Physical Society : read Oct. 28, 1898. 
Phil. Mag. S. 5. Vol. 47. No. 284. Jan. 1899. B 

2 Mr. A. Campbell on the Magnetic Fluxes in 

In some cases <I>, the total flux, was measured, and for several 
of the meters determinations of the power lost in their various 
parts were also made. 

Methods of measuring B. 

Method I. Except when the fluxes were alternating, the 
method used was the well-known way with a ballistic galvano- 
meter. A search-coil in circuit with the galvanometer either 
had the flux suddenly reversed in it, or was pulled quickly 
out of the field. For many of the experiments small and 
very thin search-coils had to be used. For instance, in one 
of the meters, the available air-gap ayrs only 1 millim. across. 
The bobbins of these small coils were made by cementing 
together one or more round microscope cover-glasses between 
larger strips of mica for the ends ; they were wound with 
from 10 to 200 turns of silk-covered copper wire of 0"075 
millim. diameter. Much thicker wire was used when it was 
desirable to have the resistance of the galvanometer-circuit 
low. The smoothness of the mica cheeks allowed the coils to 
be withdrawn from position with the necessary quickness. 

Fig. 1. 

The galvanometer was calibrated from time to time by means 
of a standard pair of coils whose mutual inductance was 
accurately known. As the measurement of small fluxes by 
the ballistic method presents no difficulty, and requires only 
ordinary instruments, further description is needless. 

Method II. When, however, the small flux is an alter- 
nating one, the voltage set up in the search-coil is more 
difficult to measure; accordingly two special methods were 
here used. In the more accurate of these two methods 
the search-coil Q (fig. 1), through which the flux was made 
to pass, instead of being in circuit with a ballistic galvano- 
meter, was joined directly to a resistance-coil R, laid along 
one set of junctions of a minute^thermopile Th, which last 
was connected w T ith a sensitive galvanometer G. The re- 
sistance-ware was of manganin (silk-covered) and was placed 
along the junctions so as to be non-inductive, and to avoid 

Meters and other Electrical Instruments. 3 

producing eddy-currents. The thermopile consistt-d of ten 
pairs of thin iron and nickel wires each 7 millim. long. 
These metals were chosen as their thermo-electric lines are 
far apart, and almost parallel to one another. 

Some years ago the writer showed (Proc. Roy. Soc. Edin. 
July 1887) that a thermopile used thus could give a fairly 
accurate measurement of the current through the resistance- 
wire, the ultimate deflexion being proportional to the square 
of this current. Hence, for a given frequency, the mean 
square of the P.D. at the terminals of the search-coil was 
proportional to the deflexion. Each time it was used the 
combination was calibrated in the following manner : — A 
measured current of 1 ampere from the alternating supply 
circuit used was passed through a non-inductive resistance of 
0*2 ohm, and the resulting P.D. of 0*2 volt was applied to 
the ends of the resistance R. From the observed deflexion 
of the galvanometer the mean square of the volts per division 
was found*. In all cases when the search-coil was in circuit 
the frequency n was observed, being measured by a frequency- 
teller. If the resistance of the search-coil be negligible, and 
if the flux follows the sine law, we have 

r 2 =:10- 8 x27r^]Sr 2 B^, 

v 2 = voltage shown by galvanometer, 

n = frequency, 

N 2 = number of turns in search- coil, 

s 2 = area of search-coil, 

B = Vuieaii square B. 

The other method, which was by means, of a telephone, 
was a rougher way, and will be described below. 

In Table I. are given some of the results obtained in the 
case of the simpler instruments, the third column giving the 
resistance of the instrument, the fourth column its full load, 
and the fifth the mean value of B at that load. As far as 
possible the positions from which the mean value of B were 
obtained were chosen so as to give an idea of the working 
flux-density, and except where otherwise stated the values 
given refer to full load. 

* The thermopile method of measuring' small voltages is now in lise 
in the German Reichsanstalt. 


Mr. A. Campbell on the Magnetic Fluxes in 
Table I. 





B (mean). 












Siemens Electrodynamometer 

Kelvin Ampere-Balance 










7X10- 5 






4 amps. 
20 amps. 
10 amps. 

50 amps. 

0-2 volt 

3 volts 

100 amps. 

18 volts 

500 amps. 
15 volts 
40 volts 


(100 volts) 
(200 volts) 
(0 - 2 amp.) 

(1 amp.) 
(1 amp.) 










(about 0-2) 

(about 10) 




Bifilar Mirror Wattmeter (after ] 
Dr. Fleming). J 

Ayrton and Mather D'Arsonval 1 
Galvanometer. J 

Weston Voltmeter 

Davies Voltmeter (Muirhead) ... 

Ayrton and Perry Magnifying! 
Spring Voltmeter. j 
Richard Recording Ammeter 

Nalder Voltmeter 

Any Tangent Galvanometer 

Kelvin Astatic Mirror Gralvano- \ 

meter. J 

Evershed Ohmmeter, Old Type . . . 

,, ,, 'New Type... 

Campbell Frequency-Teller 

Bell Telephone (double pole) 

Ayrton and Perry Variable In- ] 
ductance Standard. J 
Standard Inductance Coil "1 
(L= 0-2 henry. J 

As the numbers in the above Table throw an interesting 
light on the behaviour of many of the instruments it seems 
desirable to discuss them more fully in order. 

(1) Siemens Electrodynamometer. — Measurements taken at 
the middle and the top of the swinging coil (of 4 turns) in 
direction perpendicular to the plane of the fixed coils gave 
B = 120 and 40 respectively for the thin coil, and B=21 and 
16 for the thick. It will be seen that when the thick coil is 
used the deflecting field is quite comparable with the earth's 

This, of course, introduces an error with direct currents 
unless care is taken to place the instrument so that the 
direction of the earth's field is at right angles to that of the 
deflecting field, and in the proper sense, i. e. with the instru- 
ment looking east or west according to the direction of the 
current in the swinging coil. The above results show that 
the maximum variation at 15 amps, introduced by wrono- 
placing (viz., due to a field equal to twice that of the earth, 
or 0*36) would be about 2*5 per cent, of the mean deflecting 
field. This was verified by placing the electrodynamometer on 
a well-levelled turntable, and connecting its thick coil with a 

Meters and other Electrical Instruments. 5 

quite steady source of continuous current by means of twisted 
flexible leads. In Table II. are shown the values of the 
current indicated by the instrument when turned into various 
positions, the first column giving the direction towards which 
the front was turned in each case. 

Table 11. 



Variation from 

K . . 

. . 15-00 

W. . . 

, . 15-20 


s. . . 

, . 15-32 


E. . . 

. 15-20 


It will be seen that the extreme variation is 2*1 per cent., 
w T hich agrees (within the limits of error of the instrument) 
with the 2*5 per cent, variation deduced from the observed 
fields. For lower currents the variation is much more, being 
in the inverse ratio of the current. With the thin coil the 
field due to the fixed coil is so much stronger that the varia- 
tion is slight except at the very lowest currents. 

The thick coil had 7 turns and an area of about 46 sq. 
centim., and the total maximum flux at full load was found 
to be 1850, giving mean B==40. At full load the power 
wasted is 6 '2 w r atts and 9*3 watts for the thick and thin coils 

(2) Kelvin Balance. — The total flux through the central 
space of the coils was got by winding the search-coil round 
the supporting pillar, and taking throws by reversing the 
current. The resultant flux was about 1600 (for full load). 
By the astatic arrangement of the swinging coils the instru- 
ment is made independent of the earth's field. The self- 
inductance is about 0'0016 henry. 

(3) Bifilar Mirror Wattmeter. — This instrument has ranges 
up to 50 or 100 amperes at 10 volts and upwards. The 
numbers given refer to the fixed series coil. With direct 
currents it is clear that precautions have to be taken to elimi- 
nate the effects of the earth's field. 

(4) D'Arsonval Galvanometer. — This was a ballistic one 
(made by Paul) with a narrow swinging coil of the Ayrton- 
Mather type. The B given in the Table is that in the air- 
gap in the neighbourhood of the moving coil ; it would seem 
to be sufficiently great to be practically unaffected by the 
magnetism of the earth. Besides, as it is an instrument for 
use in a fixed position, it is only the effect of variable ex- 
ternal fields that need to be taken into account. This point 

6 Mr. A. Campbell on the Magnetic Flaxes in 

will be discussed a little further on. To get an idea of how 
the flux in the steel varies from point to point along the 
annular magnet, an experiment was made with a ring-magnet 
of rectangular section, having an air-gap as shown in fig. 2. 
A small search-coil which could only just slip along the 
magnet was moved by jerks into successive positions, and the 
corresponding changes in the flux were calculated from the 
throws on a ballistic galvanometer in circuit with the coil. 
Fig. 2 shows the result, the radial breadth of the shaded 
part being drawn proportional to the flux in the steel at each 
position. Fig. 3 is a similar diagram for an ordinary bar- 
magnet. In the ring-magnet the available air-gap flux was 
less than one third of the maximum flux at a a. 

(5) Weston Voltmeter. — It will be noticed that B in the 
air-gap of this instrument is very high, viz. 870. This might 
lead one to suppose that the earth's field would have no 
perceptible effect on its readings, but it must be remembered 
that the flux induced in a piece of iron or steel in the earth's 
field is usually very many times greater than the flux in air 
due to the earth alone. This can be easily shown by connect- 
ing a coil with a ballistic galvanometer and reversing the 
coil with regard to the earth's field first by itself and then 
with a soft iron core in it. The throws of the galvanometer 
will be enormously increased by the presence of the core. 
To find how far the earth's field affected the flux in a perma- 
nent magnet with a moderate air-gap, a coil was wound upon 
the circular one shown in fig. 2, and was connected with a 
galvanometer. The magnet was then turned round so as to 
quickly reverse the action of the earth's horizontal field upon 
it. The resulting throw on the galvanometer showed that the 
maximum B in the steel, which was about 5000, was only 
changed by 3 lines per sq. cm., i. e. by less than 0*1 per cent. 
The behaviour of the Weston magnet tallies with this, for when 
the instrument, with a steady voltage on its terminals, was 
turned round to face each point of the compass, no change 
in the reading could be detected, although the scale could be 
read to about 1 in 1000*. 

(6) Davies Voltmeter. — In this instrument one side of the 
rectangular moving coil moves in a narrow cylindrical air- 
gap between specially shaped pole-pieces of a strong perma- 
nent magnet. It gives a maximum angular deflexion of 
about 210°. 

(7) Ever shed Ammeter. — A coil of 6 turns magnetizes 
two small pieces of iron with a movable piece between them, 

* At Professor Ayrton's suggestion I have re-tested the instrument at 
the higher readings, and have detected a variation of about Ol per cent. 

Meters and other Electrical Instruments. 7 

B was measured at about the end of one of the fixed pieces 
where the movable piece faces it. 

(8) Ayrton and Perry Magnifying Spring Voltmeter. — The 
small iron tube, which is surrounded by a coil and pulled 
down by it, is 7*2 centim. long, and the iron has a section of 
about 0*12 sq. centim. The number given in the table is 
the average B in the iron for the whole length of the tube 
for a load of 15 volts, i. e. 08 of fall load. 

Fiff. 2. 

(9) Richard Recording Ammeter. — This has two solid iron 
cores (each 4 sq. centims. cross- section) round which the 
current is carried by a single turn of copper strip ; an iron 
armature carrying the pointer is attracted by these cores. B 
was measured between one core and the armature. It will 
be seen that the total flux is large ; thus a strong deflecting 
force is obtained which makes the friction of the recording 
pen of less account, but on the other hand much error from 
hvsteresis comes in. 

8 Mr. A. Campbell on the Magnetic Fluxes in 

(10) Dolivo Voltmeter. — Here a thin wire of soft iron is 
drawn down into a solenoid. The value given is for the hollow 
core of the coil (including the wire). 

(11) Nalder Voltmeter. — In this a small piece of soft iron 
moves in the magnetic field produced by a coil outside it. 
The number in the table is for the middle of the space within 
the coil. 

(12). Tangent Galvanometer. — When the deflexion =45°, 
the resultant field = earth's field -=- cos 45° 
=0*26 (in London). 

(13) Kelvin Astatic Mirror Galvanometer. — The galvano- 
meter was made very sensitive and almost unstable by means 
of the controlling magnet. The mean control field (for this 
condition) was found by measuring its sensitivity and com- 
paring it with that when the earth's field alone was used. 
The deflecting field for 1° would be less than 0*00002, and 
depends on the degree of astaticism of the suspended magnets. 

(11) Ever shed Ohmmeter — (Old type; polarized, astatic). 
The number given is a rough approximation to the B due to 
the shunt-coil alone at 100 volts. About the middle of the 
scale the field due to the series-coil would have the same value. 
(15) Ever shed Ohmmeter — (New type, with soft iron 
needle). The B given is that due to the shunt-coil alone (at 
200 volts). It is clear that the earth's field cannot introduce 
much error. In any case the errors due to external fields 
can be eliminated (as the makers direct) by reversing the 
current and taking the mean of the readings. 

(16) Campbell Frequency-Teller. — The B is measured 
between the vibrating strip and the attracting pole of the 
electromagnet. It will be seen that only a quite moderate 
field is required to throw the spring into strong vibration 
w r hen it is adjusted to the right length for resonance. 

(17) Double Pole Bell Telephone. — The diaphragm was so 
close to the poles of the permanent magnet as to form an 
almost closed magnetic circuit. A small search-coil was 
wound round one of the pole-pieces (area = 0*24 sq. centim.), 
and the diaphragm was then laid in its place. A throw of the 
galvanometer was got by pulling off both the diaphragm and 
the search-coil. Thus the flux-density given refers to the 

(18) Ay Hon and Perry Variable Standard of Self-Induc- 
tance. — With the pointer at 0*038 henry, the total flux within 
the inner wooden bobbin was about 5200 for 1 ampere. 

(19) Self-Inductance Standard (L = 0'20 henry).— This 
was a coil of 1158 turns of insulated copper wire (diameter 
1*24 inillim.), the outer diameter of the coil being 22 centim. 

Meters and other Electrical Instruments, 9 

and the height 9 centim. The B given in the table is the 
average for the whole cross-section of the coil at the middle 
of its height. The total flux corresponding to this was 17,500. 
B at the centre of this cross-section was 53. 

In addition to the instruments already discussed, experi- 
ments were also made upon a number of meters of different 
types, some being for direct, and some for alternating 
currents. As some of the types vary in construction from 
year to year, a few words of description in each case will make 
the results clearer. 

Aron Watt-hour Meter (1894 type). — Range to 50 amps. 
at 100 volts. Two pendulums, each carrying shunt-coils, 
are acted on by series-coils under them, one being accelerated 
and the other retarded. With both series and shunt-currents 
passing (at full load) the mean B between the fixed and 
movable coils was about 70. 

Frager Watt-hour Meter. — Range to 10 amps, at 100 volts. 
A meter of the " Feeler " type, in which the shunt and series 
coils form an ordinary wattmeter, whose deflexions are 
integrated at intervals, The mean B was measured as near 
the centre of the shunt-coil as possible. 

At full load B = 63, 

With shunt-current alone . B = 13, 

With series-current alone . . B = 50. 

Hookham Direct- Current Ampere-hour Meter. — Bange to 
100 amps. A small disk, surrounded by mercury which 
carries the current, is cut twice by part of the magnetic flux 
from a strong permanent magnet (of cross-section 7*5 sq. 
centim.). The disk is thus caused to turn. On the same 
spindle is a brake disk of copper (5*4 centim. diameter), 
which is also cut by a part of the flux from the same perma- 
nent magnet. Unfortunately it was not possible to take the 
meter to pieces, so the driving flux could not be measured. 
By slipping a search-coil along the permanent magnet, the 
total leakage was found to be over 26,000 lines. The total 
brake flux passes through the disk at four air-gaps, two and 
two in series, and has a value of about 5000 lines. The 
8 pole-pieces which direct the flux have cross-sections of 
1*53 centim. each. In two of them the iron near the air-gap 
is turned down so as to leave only a thin neck of about 0*12 
sq. centim. cross-section. This is supposed to increase the 
permanence of the flux. At these necks more than J of 
the total flux leaks from the iron. Whether they increase or 
diminish the permanence seems quite uncertain. The mean 
flux density at the four air-gaps was found to be about 1020. 

10 Mr. A. Campbell on the Magnetic Fluxes in 

The power spent in the meter at full load = 12*8 watts. 

The power spent in turning the spindle (at 2'2 revolutions 
per second) with full load was measured by the method 
(1) described below, and was found to be 0*016 watts*. 
Hence the efficiency of the meter as a motor = 0*125 per cent. 

Kelvin Ampere-hour Meter. — "Range to 600 amperes. In 
this a thin iron core, kept highly magnetized by a shunt- 
current, is drawn down into a solenoid which carries the main 
current. The solenoid had 6 turns (i. e. 3600 ampere-turns 
at full load), and was about 16 centim. long. At full load 
the flux density at the lower end of the solenoid was over 
250. With the shunt-current alone the total flux through the 
moving coil and core was about 330. 

Eliliu Thomson Watt-hour Meter. — Range to 50 amperes 
at 100 volts. This meter consists of a small ironiess motor, in 
which the series-current goes through the field- magnet coils 
and the shunt-current through the armature. On the arma- 
ture spindle is a brake disk of copper (13*5 centim. diameter), 
wbich passes between the narrow air-gaps of three permanent 
magnets of the shape shown in fig. 4. These magnets are 
often of different strengths, being chosen to give the proper 
brake-force for each individual meter. Their polar faces are 
about 7*5 sq. centim. The mean B in the air-gaps was about 
700. By the method of placing the magnets the greater part 
of the flux acts on the brake disk. By slipping a search-coil 
along one of the magnets it was found that the total flux at 
a was about 15000, making B about 7000. Of this flux 
nearly one half remains in the steel as far as the section at b. 

Fig. 4. 

Driving Flux. — Without the shunt-current the full load 
current gives mean B == 130 along the axis of the series-coils. 
The shunt-current at 100 volts gives a field at right angles 
1o this in which B = 10. The shunt-current also passes 
through a " compounding " coil fixed coaxially in one of the 

* After the author had measured the motor-efficiencies of several 
meters, M?\ Sidney Evershed somewhat anticipated him by announcing 
(Institution of Electrical Engineers, May 12th, 1898) one or two similar 
results, making no mention, however, of the method by which the results 
were obtained. 

Meters and other Electrical Instruments. 


series-coils for the purpose of overcoming friction at the lower 
loads. The B due to this starting coil alone is about 3. 
There is a small stray field from the series-coils perceptible at 
the brake disk, but as it is less than the ^Jq part of the field 
due to the permanent magnets its effect may be neglected. 

Effect of the Earth's Field. — Since the driving flux density 
at full load is only 130, it is clear that at the lower loads the 
rate of the meter may be considerably affected by the earth's 
field. To test this point the meter was levelled up on a turn- 
table (as in the case of the electrodynamometer already 
described), and a constant load of 4*625 amperes at 100 volts 
was kept on it. The load was measured by a Kelvin balance 
and a reflecting multicellular voltmeter. The rate of the 
meter, i. e. spindle revolutions per watt-second, was then 

(A) with alternating current : 

(B) with direct current, the earth's field helping the driving- 
field ; 

(C'i the same, with the earth's field opposing the diiving 

Table I1L shows the results of the tests. 

Table III. 

Position of Meter. 



(A) Facing East. 

, (B) „ West. 
(C) „ East, 

Alternating at 80^ 
per second. 





It will be seen from (B) and (C) that by turning the meter 
round through 180° its rate at ^ l° a ^ ^ s altered by 3*6 per 
cent., which is exactly what might have been predicted from 
the value of B given above. The rate with alternating current 
does not lie between the rates (B) and (C) as it ought, but is 
about 5 per cent, slower than (B). 

Power Spent in Meter. — The resistance of the series-coil 
= 0*0066 ohm, and that of the shunt-coil = 2030 ohms ; 
hence the power spent in heating the coils =16*5 + 4-9 = 21*1 

The power spent in actually driving the meter (at full load) 
was measured by two methods as follows : — 

(1) An arm of about 10 centim. long was attached to the 
spindle of the meter (at right angles to it). With full load 

12 Mr. A. Campbell on the Magnetic Fluxes in 

switched on, the tangential force /necessary to hold at rest 

the end of this arm was measured by the extension of a spiral 

spring which had been calibrated by known weights. 


Power (in watts) = 10 -7 ./. 27rr . — , 


r = length of arm, 

- = number of revolutions per second when the spindle 
is free to move. 

(2) In the second method the shunt-circuit was disconnected 
and joined directly to a sensitive galvanometer with a resist- 
ance of 12,000 ohms in circuit. A measured current was 
sent through the series-coils, and the spindle was turned at 
about the full rate. Thus the meter acted as a dynamo and 
gave a deflexion on the galvanometer. By watching the gal- 
vanometer-scale it was not hard to keep this deflexion steady, 
and from the known calibration of the galvanometer the 
voltage given by the armature was obtained. The total 
E.M.F. multiplied by the normal shunt-current (0*0493 amp.) 
gave the driving-power. 

Method (1) gave for the driving-power at full load 0*020 
watt, whilst method (2) gave 0*021 watt. Hence the efficiency 
of the meter as a motor is only 0*095 per cent 

Hoohham Alternating- Current Watt-hour Meter. — Eange to 
10 amperes at 100 volts. In this meter a solid iron core 
forming an almost closed magnetic circuit is magnetized by 
a shunt-coil, which latter, by reason of its large inductance, 
carries a current which lags behind the main current by 
50° to 60°. A smaller U-shaped piece magnetized by the 
main current has its poles close to the upper pole of the 
larger iron core, and one of the poles carries a copper screen. 
A small aluminium disk (8*8 centim. in diameter), partly 
between the poles, is turned by the rotary magnetic flux 
thus produced. The brake force acts on the same disk, and 
is due to a tall permanent magnet about 20 centiin. high 
(with a narrow air-gap) , as shown in fig. 5. 

Brake Flux. — The maximum flux in this magnet was 
found to be near P, and had the value 19,820, corresponding 
to B = 8800. Of this only 4400 lines ultimately cut the 
disk, much being lost by cross leakage from Q and S to the 
opposite limb. Thus less than J of the maximum flux is 
made use of. The mean B between the poles was found to 
be 650. 

and other Electrical Instruments. 


Driving Flux. — The distribution of the somewhat compli- 
cated alternating field was traced by the following method, 
which also gave rough quantitative results. A telephone T 

Fi<r. 5. 




(fig. 6) was arranged in a circuit with the search-coil F and 
the low-resistance strip H K in such a way that the strip 
could be switched out of circuit at will. One of the con- 
nexions to HF was through a sliding contact, so that the 

Fiff. 6. 

resistance of the part in the telephone-circuit could be varied 
from 0*05 ohm downwards. A current of 1 ampere was 
maintained in the strip and was from the alternating circuit 
which supplied the meter load. With the strip out of circuit 
the search-coil was moved into various positions, and the dis- 
tribution of the alternating flux observed by means of the 
sound in the telephone. To measure the flux at any position 
H K was set to such a value that the small P.D. introduced 
by it into the telephone-circuit gave a sound of the same 
loudness as that given when the search-coil was placed in the 
flux. The absolute values given by this method came out 10 

14 Ml*. A. Campbell, on the Magnetic Flaxes in 

to 15 per cent, too small ; but it proved a very convenient way 
of comparing the flux-densities at various positions. It was 
thus found that when the main current (full load) was switched 
on in additioLi to the shunt-current, the flux-density under 
the middle pole was nearly doubled. The most curious fact 
brought to light by the method was that across the air-gap of 
the permanent magnet a considerable alternating flux is in- 
duced, the value of B at that position being actually about 
5 of that between the poles of the shunt-magnet with the 
shunt-current alone. The permanent magnet forms a kind of 
secondary magnetic circuit directing around itself the alter- 
nating eddy-currents in the disk. Whether this has any 
sensible demagnetizing effect upon the permanent magnet the 
writer has not determined. 

The fhixes were measured more exactly by the method of 
the search-coil and thermopile described above. With shunt 
alone the root of mean square B was about 50 in the air-gap 
and 800 in the iron core just above the shunt-bobbin. The 
shunt-current at 100 volts (86 ~ per sec.) was found to be 
0'031 ampere. If the current followed a sine curve its maxi- 
mum value would be 0*044. When a direct current of this 
value was tried the fluxes produced were much larger than those 
with the (supposed) equivalent alternating current. This is 
partly due to the fact that the shunt-current does not follow 
the sine law, but is no doubt also due to the existence of eddy- 
currents in the iron core. That these currents even in the 
disk spread the flux anyd reduce the flux-density in the air-gap 
was shown qualitatively by placing a search-coil, connected 
with a telephone, in the air-gap of a ring electromagnet 
excited by alternating current. The sound in the telephone 
was lessened when a copper disk was held near the coil in the 
air-gap. The search-coil and thermopile method showed that 
the flux-density had been reduced by 8 per cent. 

To find how the core-flux varied with alteration of the 
voltage on the shunt-coil, by the same method the B just 
above the shunt-bobbin was measured for a number of voltages 
from 20 up to 100 volts. It will be seen from the curve in 
fig. 7 that B is very neaily proportional to the potential- 
ditference. In practice it is found that the speed of rotation 
is very nearly proportional to the voltage. 

Power spent in Meter. — To approximate to the amount of 
power spent by reason of hysteresis when the shunt-current 
alone is on, the iron was carried through a cycle by means of 
direct current of such amount as to give a maximum B nearly 
corresponding to that given with alternating current at 
100 volts. The curve obtained is shown in fig. 8. 

Meters and other Electrical Instruments, 


Fig. 7, 



Yolts on Shunt-coil. 









0-04 0- 


02 0- 

Y o 



Cc//?ff£A/T (/, 

v Shi/a/tJ 

03 ' 0-04 


16 Mr. A. Campbell on the Magnetic Fluxes in 

From this curve it is not possible to get the exact value of 
the hysteresis loss, as the iron core is not a uniformly mag- 
netized closed circuit. To get an idea, however, of the amount 
of the power wasted by hysteresis, let us suppose that the 
magnetic circuit is equivalent to a uniform iron ring of the 
same cross-section as the core of the shant-coil, uniformly 
wound with the same number of turns n x as the shunt-coil * 
(carrying the same current) , and traversed by a flux equal to 
that at the part of the meter-core for which the curve in fig. 6 
was taken. 

Let c = current at any moment, 

anci s = section of ring. 


Hysteresis-loss in joules per cycle 

= §cdv = 10"% jc^ = W-\s§cdB 
= lO" -8 ^ x (area of curve) 
= lO" 8 x 1200 X 3-7 x area 
= 0-00114. 
.*. at 86^~ per second, 

Power spent = S6x 0*00114 = 0'098 watt. 

It was found by direct measurement that the actual power 
spent in the meter (with the shunt-current alone) was far 
larger than this. Accordingly measurements were made of 
the rise of temperature of the iron core by means of an iron- 
nickel thermopile. Three junctions of the pile were bound 
against the iron, which had its surface well insulated with 
paint. A pad of wadding was tied over the spot, and the 
thermopile was connected with a suitable galvanometer. The 
shunt-current was switched on for 120 seconds, the deflexion 
being read at intervals, and on breaking the current the 
cooling was observed for several minutes. The thermopile 
and galvanometer were calibrated with a known difference of 
temperature, and the curves of heating and cooling were 
drawn. The curve of heating was then corrected by means 
of the other curve, and thus the heating of the iron (cor- 
rected for cooling) was found. 

At a spot just above the shunt-bobbin the corrected rise of 
temperature w r as o, 87 C, whilst near the air-gap it was only 
about half of this. Taking account of this last fact, we may 

* The exact value of n x was not known, but the value (1200) used 
was estimated from the resistance and gauge of the shunt-wire and the 
size of the coil. 

Meters and other Electrical Instruments. 17 

take that a volume of about 61 c.c. was raised in temperature 

by 0°'87 C. 

rower spent = 4*2 watts, 

where <r = sp. heat of the iron, 

V = volume „ „ 

D = density „ ,, 

^T= temperature-rise (corrected), 

t = time in seconds; 
therefore from the results above, 

Power = 1*58 watts. 

Since the hysteresis-loss is about 0*10 watt, it will be seen 
that the eddy-current loss = 1*48 watts. 

The power lost by eddy-currents in the disk was similarly 
measured, and was found to be about 0*02 watt. 

In the shunt-coil the C' 2 R loss =0*57 watt ; and hence the 
total copper and iron losses =2*17 watts. 

Direct Measurement of Total Power. — The total power 
given to the meter was measured by the three-voltmeter 
method, in which was used a reflecting electrostatic voltmeter 
accurate to about 1 in 1000 at all the points of the scale 
required. The result obtained was 2*06 watts, which agrees 
fairly well with the total watts shown by the other methods. 

Driving- Power. — The driving-power at full load was also 
measured by the method of the spring-balance described 
above ; it was found to be 0*00073 watts. Taking account 
of the series-coil (whose resistance was 0*009 ohm) the total 
power spent at full load is over 3 watts ; and hence 

Motor efficiency = 0*024 per cent. 

Scheefer Alternating- Current Watt-hour Meter. — Range to 
10 amperes at 200 volts. In this meter a pile of E-shaped 
iron stampings has the series-coil on one outer limb and the 
shunt-coil on the other. An aluminium cylinder is turned 
by the rotary field thus produced, and on the same spindle is 
a brake disk with a single magnet exactly like those in the 
Elihu Thomson meter. 

The mean B between the poles of this magnet was about 
480. With the shunt-current alone the -v/mean B 2 between 
the shunt-pole and the driving cylinder was 570. The total 
power spent with the shunt alone on is 11*5 watts, of which 
7*1 watts are due to C 2 R loss. 

Shallenberger Alternating- Current Ampere-hour Meter. — ■ 
Range to 20 amperes. A series-coil (of about 50 turns) and 

Phil. Mag, S. 5. Vol. 47. No. 284. Jan. 1899. C 

18 Magnetic Fluxes in Meters and other Electrical Instruments. 

a small short-circuited secondary coil, with their axes at an 
angle of about 45°, produce a rotary field by which is turned 
a small disk with a soft iron rim. The brake-force is obtained 
by air-friction on four aluminium vanes. 

By the search-coil and thermopile method it was found that 
inside the series-coil 

VmeanB 2 =100. 

Power spent in Meter. — The resistance of the series-coil 
was 0'025 ohm ; hence the power spent in it =10*0 watts. 

The power spent in the copper stampings which form the 
secondary coil was found by measuring their rise of tem- 
perature with a small copper-iron junction. This rise (at 
full load) was found to be about o, 33 C. per minute, the 
cooling being negligible. The volume of copper was about 
21 '4 c.c. ; whence the power spent = 0'40 watt. 

The driving-power was found to be 0*0069 watt ; therefore 
the motor efficiency =0'066 per cent. 

Current in Secondary Coil. — The current in the short- 
circuited secondary coil could not be measured directly. 
Calculating from the dimensions of the coil, however, the 
resistance was foimd to be 8*5 x 10 -6 ohm. From this and 
the value of the power (0*40 watt) we find that the secondary 
current attains the extraordinary value of 220 amperes. 

For the sake of comparison some of the above results are 
collected in Table IV. 

Table IV. 


Driving B. 

Brake B. 




Elihu Thomson 











per cent. 




Hookham (direct curr.) ... 
Hookham (alternat. curr.) 

In conclusion, it will be noticed that in motor meters the 
driving-flux density is of the order 100 and the brake B from 
500 to 1000 ; also that the motor efficiencies are all very 
small, particularly in the case of the alternating-current 
meters. In all of them the greater part of the power taken 
is spent in heating conductors (either by eddy-currents 
or otherwise). If a small fraction of this wasted energy 
could be employed to overcome with certainty the friction at 
the lowest loads, a great advantage would be gained thereby. 

June 7th, 

[ 19 ] 

II. A Resonance Method of Measuring En^gy dissipated 
in Condensers. By Edwakd B. Rosa and Akthur W. 
Smith *. 

THAT the dielectric of a condenser becomes warmed when 
an alternating electromotive force is applied to the 
terminals of the condenser has long been known, and the 
study of this heating effect has been undertaken by numerous 
observers. Kleiner f used a thermo-electric couple imbedded 
in the dielectric to measure the rise of temperature, and noted 
a considerable heating effect in ebonite, gutta percha, glass, 
and mica, but none at all in paraffin and kolophonium. He 
reports that in spite of all attempts by variations of the con- 
ditions of the experiment, no heating could be detected 
in these two last-named substances. On the other hand, 
Boucherot % has made paraffin-paper condensers for use on the 
3200-volt commercial circuits of Paris, some of which became 
so hot in use that they were obliged to be cut out. Boucherot 
says of the heating effect that if a condenser rises as much as 
30° C, it should be rejected. That this is good advice is 
evident from the fact that paraffin melts at 54° C. ; and hence 
when 30° C. above the temperature of a summer's day (say, 
25° C. or 77° F.), the paper would be floating in melted 

Bedell, Ballantyne, and Williamson § report experiments 
upon a paraffin-paper condenser of 1*5 microfarad capacity, 
the efficiency of which was found to be 95*6 °/ , or 4*4 °/ 
lost in heat. The loss was determined by a three-voltmeter 
method, similarly to measurements on a transformer. It was 
put upon a 500-volt circuit at a frequency of 160, and the 
current was therefore about 07 ampere, the apparent watts 
about 350, and the heating effect 15*4 watts. The tempe- 
rature rose one degree per hour. 

Threlfall |] reports a test on a paraffin-paper condenser of 
his own construction which had a capacity of 0'123 micro- 
farad, and on a circuit of 3000 volts and a frequency of 60 
the rise of temperature was less than one-fifth of a degree per 
hour. The apparent watts would be about the same as in the 
experiment of Bedell, Ballantyne, and Williamson, although 
the capacity was only one-twelfth as much. Threlfall concludes, 
apparently, that since his condenser had only one-twelfth the 

* Communicated by the Authors. 

t Wied. Ann. vol. 1. p. 138. 

% LEclairage Electrique, Feb. 12, 1898. 

§ ' Physical Review,' October 1893, vol. i. p. 81 

|| ' Physical Review/ vol. iv. p. 458 (1897). 


20 Messrs. E. B. Rosa and A. W. Smith on a Resonance 

electric capacity, it also had only one -twelfth the capacity 
for heat of the other condenser ; and since the temperature 
rose five times more slowly, therefore that the percentage loss 
of energy was only one-sixtieth as great, that is, £$ of 3 %, 
or 0*05 °/ , giving his condenser an efficiency of 99*95 °/ ! 
The reasoning is, however, quite inconclusive, for nothing is 
said concerning the thickness of the paraffin-paper dielectric 
and the volume of the condenser. Suppose the dielectric of 
the smaller condenser to be 0'0129 inch thick, which is three 
times the thickness of the other. (It might have been as 
thick as that, seeing it sustained for a considerable period an 
alternating voltage of 3000.) Then the volume per unit of 
capacity would be nine times as great as the other, and the 
heat required to raise its temperature as rapidly would be 
nine times as great, assuming the capacity for heat and rate 
of radiation the same for both. The loss would then have 
been 0*45 % instead of 005 %. This illustrates how entirely 
inconclusive any determination merely of rise of temperature 
is, unless all such circumstances as heat capacity, rate of 
cooling, thickness of dielectric (or intensity of the electro- 
static induction) are specified. 

The enormous discrepancy between the results above re- 
ferred to as to the quantity of the heating effect in condensers, 
and the almost entire lack of precise statements as to its 
numerical value, led us to undertake, more than a year ago, 
to measure this energy loss in such a way that it could be 
expressed absolutely. 

We proposed to measure by means of a wattmeter the 
energy dissipated in a condenser when it is subjected to an 
alternating electromotive force. In order that the frequency 
of charge and discharge be perfectly definite, the electromotive 
force should be a simple harmonic one, that is, the upper har- 
monics of the fundamental should be absent. This is most 
easily effected by inserting in the circuit in series with the 
condenser a coil of wire having large self-induction, but 
without an iron core. The variable permeability of the iron 
will give rise to upper harmonics, especially if the magnetic 
induction of the core attains large values ; and hence a coil 
without an iron core is necessary. Moreover, if the self- 
induction is not large enough it will reinforce some of the 
upper harmonics instead of quenching them; and the presence 
of the coil will be detrimental rather than beneficial. The 
best value of the self-induction of the coil is such that the 
fundamental is reinforced to a maximum degree; in other 
words, it is that value of L given by the equation 


Method of Measuring Energy dissipated in Condensers. 21 

where t is the period of the fundamental component of the 
impressed electromotive force and the capacity of the 

There is another practical advantage resulting from this 
arrangement, namely, that the resulting resonance raises the 
electromotive force at the terminals of the condenser very 
greatly, saving the necessity of raising the voltage by trans- 
formers. And a third advantage now appears in the fact that 
the wattmeter may be inserted across the low-voltage supply 
wires to measure the total power expended upon coil and 
condenser. Then subtracting the IV loss of the coil, the 
remainder will be the power expended upon the condenser. 
This supposes, of course, that there is no iron core and no 
eddy-current loss in the copper coil itself. 

The Resonance Method, 

Fig. 1 shows the connexions for this method with the coil 
in series with the condenser. M, N are the low-voltage 
supply-wires of an alternating circuit, S is an adjustable re- 
sistance, E is a dynamometer, F a wattmeter, R a non- 
inductive resistance in shunt with the coil and condenser, 

Fig. 1. 





r * 

p ■ 


1 B 

V V V 




joining the points A and D. The wattmeter therefore mea- 
sures the power expended between the points A and D, 
including the Vr w loss in the fixed coil of the wattmeter, but 
not including the i 2 J& loss in the shunt-resistance R. The 


•22 Messrs. E. B. Rosa and A. W. Smith on a Resonance 

energy expended in the resonance -coil and the fixed coil of 
the wattmeter, I 2 (r c -\-r w ), subtracted from the total power 
measured leaves the condenser loss. This remainder is found 
to be also proportional to the square of the current. Hence 
we may write it I 2 r s , and r s is the u equivalent resistance " of 
the condenser, which it is desired to find. This does not 
indicate the nature of the process by which energy is dissi- 
pated in a condenser, but simply that for a given condenser 
made of a given dielectric at a given temperature and fre- 
quency, the heating effect is the same as though there were a 
certain resistance r 8 in series with a perfect condenser of the 
same capacity. For the same dielectric r s changes with 
changes in the temperature or the frequency; and for another 
dielectric with the same capacity, temperature, and frequency 
r s would be different. 

If £ = 27r\/LO, there is complete resonance; t= , and 

1 . 1 n 

p = 27rn. Therefore - = n/LC, or 0=-^. That is, for 
p p 2 h ' 

complete resonance the capacity is inversely proportional to 
the square of the frequency for a given self-induction. If 

Fig. 2. 

B b 

A a 

a c D 

the frequency is fixed, either the capacity or the induct nee 
may be varied until the current is a maximum ; but if the 
frequency can be varied, the maximum resonance may be 
attained without varying C or L. 

Method of Measuring Energy dissipated in Condensers, 23 

In fig. 2, let Aa = i*c, the ohmic resistance of the coil; 

aB = pL, the reactance of the coil ; 

Bb=r s , the equivalent resistance of the con- 

-* denser ; 

bG = -7s, the condensance of the condenser ; 

CD = r w , the resistance of the fixed coil of the 
wattmeter, its induction being ne- 
gligible ; 

AD = the resultant impedance of the circuit. 

Then if pL= p-, the reactance is equal and opposite to the 

condensance (fig. 2 c) , the resonance is complete, and the 
impressed electromotive force e is expended in overcoming 

the resultant resistance AD=r c + r s +r. and 1= — - - — . 

r c + r s + r w 

lines represent 

In fig. 3, similar to fig. 2 c, the several 
electromotive forces. Aa = Ir c = e c = that part 
of the electromotive force expended in over- 
coming the ohmic resistance of the coil ; simi- 
larly, B# = e s , OD = e w . Ba is the electromo- 
tive force (due to resonance) which overcomes 
the reactance, and bG is the electromotive force 
which overcomes the condensance. The po- 
tential of the point B varies through a wide 
range; whereas the points A, C, and D, and 
the instruments suffer only small changes of 

Fig. 3. 


Advantages of the Method. 

Herein lies one of the chief advantages of 
the method, that voltages below a hundred 
have to be dealt with at the instruments, 
whereas upon the condenser there may be an 
active electromotive force of several thousand 
volts. The noninductive resistance E, is at 
most a few hundred ohms. On the contrary, if the shunt- 
resistance were applied directly at the terminals of the con- 
denser, it would necessarily be several thousands of ohms, it 
must be capable of carrying the entire shunt-current of the 
wattmeter, it must be strictly non-inductive, and be of 
known value — conditions difficult to fulfil. In the resonance 
method a small inductance in the shunt-resistance or the 
movable coil of the wattmeter produces no appreciable error; 
whereas in the simple wattmeter method it produces a large 


24 Messrs. E. B. Rosa and A. W. Smith on a Resonance 

To illustrate this point let us take a special case. Suppose 
the resistance R is 500 ohms and the inductance of the shunt- 
circuit, including both the resistance R and the movable coil 
of the wattmeter, is 0'003 henry, %7rn being 800. Then 
^L = 2-4 and tan (^ = 0-0048, ^ = 16' 30", the angle of 
lag of the shunt-current behind the electromotive force. 
Suppose the true angle of the condenser-current, <£ 2 (fig. 4), is 
89° 40' ahead of the electro- 
motive force. Then the dif- 
ference of phase of the two 
currents in the wattmeter will 
be = ^ + 03 = 89° 56' 30", 
and the power factor, cos cf>, 
of the expression watts = 
EI cos^will be cos 89°56 / 30" 
instead of cos 89° 40', that is 
•00102 instead of '00582; thus 
the wattmeter would indicate 
that the power absorbed in the 
condenser was only about one- 
sixth of what it really is. If 
the lag of the shunt-current 
were more than 20 x (a not 
improbable value in many 

cases), the deflexion of the wattmeter would be negative ! 
It is possible that this explains why it has often been claimed 
that the loss in certain condensers is too small to be measured 
by a wattmeter. For example, Swinburne*, speaking of 
some of his own condensers, says " a condenser that takes 
2000 volts and 10 amperes has a loss that is too low to measure 
— that is to say, it is less than 5 or 10 watts." 

On the other hand, if by means of a resonance-coil the 
current and electromotive force have been brought very 
nearly, if not exactly, into phase, any small lag of the shunt- 
current will make no appreciable error. Thus, the cosine of 
10° is -9848, and of 10° 16' 30" is '9840, a difference of less 
than one part in a thousand. 

In order to determine the precise values of r e and r w a 
Wheatstone bridge is joined to A D and the condenser short- 
circuited, so that the resonance-coil and the fixed coil of the 
wattmeter and the lead-wires form the fourth arm of the 
bridge. The resistance is then quickly measured just after 
the wattmeter has been read and the alternating circuit broken, 
and changes due to temperature are included. 

* < Electrician/ Jan. 1 (1892). 

Method of Measuring Energy dissipated in Condensers. 
Difficulty of the Method. 

We have seen that the presence of a resonance-coil in series 
with the condenser (1) quenches, to a large extent at least, the 
upper harmonics, (2) raises the voltage upon the condenser, 
thus avoiding transforming up, (3) enables measurements to 
be made more safely and more conveniently upon a low voltage, 
and (4) transfers the wattmeter problem from the most un- 
favourable case (where the angle of phase-difference is nearly 
90°) to the most favourable case where the current and 
electromotive force are nearly in phase. There is, however, 
one serious difficulty in the method. If the resonance- coil is 
made of small wire, it has a great resistance, and of the total 
power measured only a small part is expended on the con- 
denser. Thus the condenser loss is the difference between 
two relatively large quantities, and cannot be determined as 
accurately as would be desired. If, on the other hand, a 
large coil of larger wire is used so that its resistance is small, 
there will be eddy-currents in the copper of the coil, and the 
power expended on the coil will be greater than 1V C . This 
excess will go into the remainder as condenser loss, and may 
give rise to a considerable error. If the wire is of large cross- 
section, but stranded, so that its resistance is small and the 
eddy-currents negligible, then a large coil will have a large 
inductance, and no difficulty appears. The method is then 
accurate as well as quick and convenient. 

The Resonance Ratio, 

As the condenser is alternately charged and discharged 
energy is handed to and fro between the coil and the condenser. 
When the condenser is charged to its maximum extent the 
current is zero and all the energy is potential and residing in 
the condenser. A quarter of a period later the condenser is 
discharged and the current is a maximum ; the energy is now 
kinetic, and resides in the magnetic field of the resonance- 
coil. At other instants the energy is partly potential (in the 
condenser) and partly kinetic (in the coil). As this transfer 
of energy to and fro continues, the dynamo supplying the 
current furnishes just enough energy to make good the losses, 
that is, the heating effect in the wires and the dielectric of the 
condenser. The losses due to electromagnetic radiation and 
mechanical vibrations are usually negligible. 

For the condenser alone, 

i= E 

\A + <y 


26 Messrs. E. B. Rosa and A. W. Smith on a Resonance 

I and E being the square root of the mean square values as 
indicated by an electrodynamometer and electrometer. 
For the combined circuit 

1 = 

\/(r w + r e + r s y+(p^-^j 

where e is the small impressed electromotive force and the 
denominator is the combined impedance of the circuit. 
For complete resonance, 

jt)L= p-, and hence 1= 

Cp' r w + r e + r a 



E V c C 2 p 2 Impedance of the condenser ^ 

— = — = — - — rfn—i rr =±tesonance ratio. 

e r c + r s + r w 1 otal resistance 

In one case e was 50 volts and E was 2250, giving a reso- 
nance ratio of 45. The impedance was 51 ohms, r +r w 


was *38, r s was '72. Hence ' =46*4, agreeing very 

nearly with the ratio of the voltages. In this case the coil 
was of large wire (No. 5 B & S) , and had considerable eddy- 
current loss. Hence the value '72 for r s was too large, and 
the degree of resonance was lower than it would have been in 
the absence of eddy-currents. In another case, using a coil of 
No. 10 wire, the impressed electromotive force was 29*5 volts, 
the voltage on the coil or the condenser was 1808, and the 

resonance ratio — was therefore 61*9. 

The Resonance- Coll in Parallel. 

A second arrangement of the resonance-coil is to put it in 
parallel with the condenser (fig. 5), and impress upon both a 
high electromotive force. Each of the two parallel circuits 
from B to C takes its own current, independently of the 
other, but being nearly opposite in phase they nearly cancel 
each other in the supply wires. Hence a small transformer 
is sufficient to supply the small current needed, although 
without the resonance-coil a large transformer would be 
necessary. If, as before, 

pL= k' 

Method of Measuring Energy dissipated in Condensers. 27 

there is complete resonance. The two parallel circuits having 
the same impedance take the same current ; one current is 
nearly 90° ahead in phase of the electromotive force (see 

Fig. 5. 






curve 2, fig. 6), the other is nearly 90° behind (curve 3), the 
sum of the two being relatively very small (curve 4) and in 
phase with the impressed electromotive force (curve 1). The 
shunt resistance must be applied at the high voltage terminals, 
but as a small amount of self-induction produces no appre- 
ciable error in the wattmeter, the movable coil may be long- 
enough to make the wattmeter quite sensitive, and so a quite 
small shunt current may be used. This requires a larger 
resistance, but with much smaller carrying capacity, since a 
much smaller shunt current will suffice than when the main 
current differs largely in phase from the shunt current. 

To illustrate this point, suppose as before that for a given 
condenser the angle of advance of the current is 89° 40'. 
The power factor, cos <j>, in the expression watts = E I cos <£ 
is in this case '00582. If now a resonance-coil be placed in 
parallel with the condenser and the current in the fixed coil 
of the wattmeter brought into phase with the electromotive 
force, then cos <£ = 1. To get a certain deflexion of the watt- 
meter, therefore, we must have the product of the two currents 
in the wattmeter nearly 200 times as great in the first case 

28 Messrs. E. B. Rosa and A. W. Smith on a Resonance 

Fig. 6. 

Method of Measuring Energy dissipated in Condensers. 29 

as in the second, and this requires a relatively large shunt 

A modification of the method, if a second small transformer 
is available, is to transform down again to a low voltage, and 
put the shunt circuit of the wattmeter on the low voltage 
secondary of this second transformer. The current will now 
be almost exactly opposite in phase to the high electromotive 
force at the terminals of the condenser, and by interchanging 
the terminals the wattmeter deflexion will be the same as 
before, if the shunt resistance is reduced in the ratio of trans- 
formation. The currents in the two coils of the wattmeter 
are so nearly in phase with one another that a small change 
in the phase of the shunt current will produce no appreciable 

The Efficiency of a Condenser. 
Having thus determined the energy, w, dissipated in a 

condenser, by wattmeter measurements, we 
readily find r s , the equivalent resistance of the 
condenser, from the expression 

10= IV. 

The ratio of the equivalent resistance to 

c P 

is cot $ (fig. 7) , <f> being the angle of advance 

of the current ahead of the electromotive force. 
It remains to calculate the efficiency of a con- 

In fig. 8 I is the current flowing into and 
out of the condenser, assuming both current 
and electromotive force to be simply harmonic. 
The dotted curve is the power curve. 

^=E X sin pt j 

where 0=the instantaneous E.M.F. acting on 
the condenser, and E x is its maximum value. 

E x 

sin (pt-{-cj)) = I 1 sin {pt + <f>), 

Fig. 7. 


£?' =E 1 I 1 sin pt sin (pt + </>) 

= EJi [sin 2 pt cos $ + sin pt cos pt sin <£] . 

The area of the power curve for one half-period, that is the 
area of the positive loop from B to C, minus the area of the 

30 Messrs. E. B. Rosa and A. W. Smith on a Resonance 

Fig. 8. 


Jfethod of Measuring Energy dissipated in Condensers. 31 
negative loop from C to D, is given by the integral 

I ^'^=E 1 I 1 cos^>l sin 2 p£ ^ + E^ sin </>| sinptcosptdt, 

EJi jFpt sin2»rr EJj . ,r _ 1» 

1 EJj , 

and this is the work done in — of a second. 

Hence for one second the work done, and therefore the 
power, is the well-known expression 

P^Ii cos </> = EI cos (E = Ei t/I), 

E and I being the effective values of the electromotive force 
and current. 

The area of the positive loop, which represents the work 
done upon the condenser in charging it, is 

and the area of the negative loop, which represents the work 
done by the condenser in its discharge, is the value of the 
same integral between the limits tt— <£, and ir, for pt. 

The efficiency is the ratio of the work done by the condenser in 
its discharge to the work done upon the condenser in charging 

This is the gross efficiency, r\. 

mi- . oqo Area of the Negative loop _ l/^E 1 I 1 x "48638 _ Q , „- 

Taking <£ = 89°, V = Area f Positive loop " l/pE&x '51378 ~ 94 6 ? % 

i. *=87°,? = „ = S= 84 * 75 % 

„ (£ = 45°, *? = = = 6-38 o/ 

Having found the angle <£ by the wattmeter measurements 
the gross efficiency of the condenser may be found from the 
above equations, or taken from fig. 9, which is drawn from 

32 Messrs. E. B. Rosa and A. W. Smith on a Resonance 

Fig. 9. 








\ \ * 



\ n 

\ * 

\ c> 

\ % 


\ <* 




\ v 



85° 80° 

7o z 

70 c 

65° 60° 55° 
Angle (p. 


45 c 


35° 30 c 

Second Definition of Efficiency. — Net Efficiency. 

Regarding the condenser as an instrument for storing 
electrical energy, and one in which a certain amount of 
energy is dissipated in the process, we may define the per- 
centage of loss as the ratio of the energy dissipated to the 
energy stored. The efficiency is then unity minus the loss, 
or, in per cent., the efficiency is 100 minus the per cent, of 
energy dissipated. This we may call the net efficiency, and 
represent it by e. Then 

Energy Stored — Energy Dissipated 1 Energy Dissipated 
Energy Stored ; Energy Stored 

In fig. 10, 
I 1 r g = E 1 cos <£ = active electromotive force ; 

^-=E 1 sin <£ = wattless electromotive force, or the E.M.F. 

which charges the condenser. 
The maximum charge of the condenser is 
CE X sin <£, 

6 = 

Method of Measuring Energy dissipated in Condensers. 33 

and its energy Fig. 10. 

W=iOB 1 2 sin 2 0. r rs 

The expression for the power is 

JliEj cos (j>, 

But , ^^ . , 

I 1 = CE 1 sm</>.p. 

Hence the energy dissipated per second is 

•JCEi 2 sin (j> cos (j> .p, 

and the energy dissipated per half-period (that 
is during the time of a single charge and dis- 
charge) is 

zy = iCE 1 2 sin <j) cos cj> . it. 

The relative loss is therefore 

^ = 7TCOt(/>, 

and the net efficiency is 

6=1 — TTCOt (f>. 


= 89°, 6 = 94-52%, 

= 88°, e=89-03 / , 

<£ = 87°, 6 = 83-54 0/0, 
= 72° 20' 30", e=0. 

The net efficiency, e, is therefore slightly less than the 
gross efficiency, rj, for values of cf> nearly 90° ; but, as the 
angle <\) diminishes, e falls rapidly below rj, and for 
= 72° 20' 30" the energy dissipated is equal to the energy 
stored,' and the net efficiency is therefore zero, while the 
gross efficiency is about 38 per cent. For greater angles of 
lag the loss is greater than the maximum energy stored, and 
e becomes negative (see fig. 9). 

For ordinary cases the angle <j) is greater than 88°, and e 
and 7/ are nearly equal. Since the wattmeter method gives 
directly the value of (f> it is much easier to express the value 
of the net efficiency e (namely, 1 — 7r cot <£) than the value 
of 7]. For small values of cot <£ this is sufficiently exact to 
write it e = 1 — it cos (j>. 

Suppose that instead of assuming the effective resistance r s 
of the condenser to be in series with the condenser, as we 
have done in figs. 2, 3, 7, and 10, we consider that it is in 

Phil Mag. S. 5. Vol. 47. No. 284. Jan. 1899. D 

34 Messrs. E. B. Eosa and A. W. Smith on a Resonance 

parallel, as in fig, 11. Of course there is a slight leakage 
current in every case, if the resistance of the dielectric is not 
infinite. Boucherot* says of his paraffin-paper condensers 

Fig. 11. 


that the " heating is chiefly due to the Joule 
effect, that is, to leakage current; the action 
of dielectric hysteresis, if it exists at all, 
being very slight." We shall give reasons in 
a subsequent paper for believiug that this is 
seldom, if ever, true of good condensers, but 
at present let us assume it to be true. Then 
the condenser current is 90° ahead of, and the 
leakage current in phase with, the impressed 
electromotive force. I being the total cur- 
rent, the condenser current is I sin <f> and the 
leakage current is I cos <£. The energy 
stored is 

W = iCE 1 2 , 

and the energy dissipated per second is 

JExIj cos <£, 

or per half-period 

™ = 4n ElIlC0S( ^ 
The maximum condenser current, I x sin <£, =pCE 1 . 


_ E 1 2 .joCcos<j> _ 1 
4n . sin (f> ~ 2 

CExV cot (f>. 


as before. 

V 6=1 — ™.= 1 — 7T COt (py 

Referring to fig. 13, we can derive anew the value of the 
net efficiency. Curve 1 is the electromotive force, curve 2 
is the current, in advance in phase by the angle (/>, nearly 
90° ; curve 3 is the power, the positive and larger loop being 
the work done on the condenser, and the negative and smaller 

* VEclairage Electrique, Feb. 12, 1898. 

Method of Measuring Energy dissipated in Condensers. 35 

Fig. 13. 


36 Messrs. E. B. Rosa and A. W. Smith on a Resonance 

loop being the work done by the condenser upon its discharge. 
Equation°(l), p. 31, shows" the area of this power-curve to 
consist of two terms, the coefficient of one containing I cos 
and of the other I sin <£>. JsTow Icos<£ is the component of 
the total current which is in phase with the E.M.F., and is 
represented by curve 5. lsin<£ is the condenser current. 
90° ahead of 'the E.M.F., and is represented by curve 4. 
The power-curve for 4 is 6 ; its positive and negative loops 
are equal, and it is the power-curve for a perfect condenser. 
The power-curve for 5 is 7, and is the total work done, or 
the energy dissipated. One loop of 6 is energy stored, W, one 

loop of 7 is energy dissipated, w, and the ratio ^ is the 

relative loss, or 1 — v^ is tne net efficiency. 

The area of one loop of 7 is 

= EJ^ r| _ ^ipp j - = gj, cos f* =Energy pitted. 

The area of one loop of 6 is 

W= EA sin <f> |- cos jptl^t EJl * in * . 2=Energy Stored. 

.'. 6=1— ^ =1 — 7rcot 0, as before. 

If the equivalent resistance of the condenser is taken to be 
a series resistance then we have E cos <j) for the active E.M.F., 
E sin cj> for the condenser E.M.F., and the same result follows. 

Example of the Resonance Method. 

The condenser used was one which we had made ourselves, 
and consisted of paraffined paper and tinfoil. The paper was 
12x17 centim. and '0038 centim. thick; the tinfoil was 
•0025 centim. thick, and its effective area was 10 x 15 centim. 
approximately. The paper and tinfoil were piled up dry and 
clamped between brass plates. It was then placed in melted 
paraffin and maintained for some hours at 100° to 150° C. 
This condenser then had a volume of about 300 cub. centim. 
and a capacity of about *8 microfarad. 

The resonance-coil consisted of 3000 metres of No. 10 wire 
(B and S gauge, '259 centim. diameter) wound into a coil of 
40 centim. internal diameter, 56 centim. external diameter, 
and 17 centim. axial length. Its resistance was about 10 ohms 
and its inductance 1*60 henrys. This coil was wound in ten 

Method of Measuring Energy dissipated in Condensers. 37 

sections, so that by choosing different sections or combina- 
tions of sections, a wide range of inductance could be secured. 
In this particular case the entire coil was used. The fre- 
quency of the alternating electromotive force was varied by 
varying the speed of the generator, complete resonance being 
attained at a speed of 2175, for which the frequency is 145. 
The current was 1*20 amperes, the resistance r c + r w was 
9-82 ohms, I*{r a + rJ = 14'15 watts. The wattmeter gave a 
deflexion of 188, corresponding to 37*6 watts. This leaves 
23'45 watts for condenser loss, or EI cos <£. 

E being 1808, 1 = 1-20, EI = 2169, and cos 0= |^ ='0108; 

7r cot <j> = 3*39 per cent., e= 96'61 per cent. 

The quantity of the eddy-current loss in the coil does not 
of course appear. From subsequent experiments we became 
satisfied that it w r as large enough to cause a serious error in 
the above value of the condenser loss. Hence we shall not 
give any of the other values found using this coil. The 
results obtained over a range of from 400 to 2250 volts 
showed that the loss is sensibly proportional to the square of 
the electromotive force. This conclusion is not seriously 
affected by the presence of eddy-currents, since the latter are 
themselves proportional to the square of the E.M.F. and yet 
are not large enough to swamp the condenser loss. 

We therefore wound up a coil of nearly 2000 metres No. 14 
wire (B and S gauge, diarn. '160 centim.), in 41 layers of 
45 turns each, external diameter of the coil being 37 centim. 
The eddy-current loss in this coil is less, owing to the smaller 
diameter of the wire and the smaller quantity; a subsequent 
measurement by an independent method gave 3*2 percent, as 
the increase of the effective resistance by the eddy-currents 
at a frequency of 120. At a lower frequency it would, of 
course, be less. Its use will therefore illustrate the method 
and give a fairly accurate value of the condenser efficiency. 

Measurements on Beeswax and Rosin Condensers. 

We give below a series of measurements on the efficiency 
of a set of commercial condensers made of tinfoil and paper, 
the latter being saturated with melted beeswax and rosin. 
We understand that they are piled up dry, as we have done 
with condensers made in our laboratory, and while immersed 
in the melted beeswax and rosin placed in a receiver from 
which the air is exhausted, to free them from air and moisture. 
With the details of the process we are not, however, 



>>-2 -©- 

° £ ~ *= 









£1 8.1 5 















a .s JL 


C o 









t M II 
























■©-^ — 

















-g „ 






























e sT 

C o 
c 2 














T— 1 




'o S 











-1 ~ 

5 § '!> 



i— 1 
































5_ as 


































o -H 




-1 a 

- N 









•— i 













1— 1 


i— i 




? ^ 












1— t 














8 "2 © * 

c S'S ^ • 


















c u 

D — E = i— " 





T— 1 




E| c =-i : : 


— H 






s i sa 










, 2 •- 
a, o © 

L = 




S = c 

" CO 







EH ^Q 

i - 

i— 1 







A Method of Measuring Energy dissipated in Condensers, 39 

acquainted, and cannot say whether it is something in the 
method of manufacture or the nature of the dielectric which 
makes the dissipation of energy so large ; we presume, how- 
ever, that it is the latter. There were six condensers joined 
together, each being a solid slab of about 11 centim. x 15 cen- 
tim. x 1*5 centim., thus having a volume of about 250 cub. 
centim., and a capacity of one third of a microfarad. 

The six slabs were placed on a table, joined together in 
parallel, and in series with the resonance-coil (which was at a 
distance from them and from the measuring instruments), 
loosely covered with a woollen cloth, and coil and condenser 
subjected to an alternating electromotive force of about 
50 volts, and a frequency of 120. No effort was made to 
secure the maximum degree of resonance, and the voltage on 
the condensers was found to be about 900. In a short time 
the temperature of the condensers had risen to 30°, as indi- 
cated by a thermometer inserted between two of them, and 
the first set of readings was taken. The loss of energy in 
the condensers was greater than it had been at lower tem- 
peratures, and continued to increase as the temperature rose. 

At the same time, owing to this increase in the equivalent 
resistance of the condenser, the resonance ratio decreased and 
the current and voltage on the condenser decreased. The 
loss at 36° C. is 50 per cent, greater than at 30° C, and is 
approaching a maximum. At 39° it is 9*5 per cent., and the 
fourth reading at sensibly the same temperature (but which 
doubtless was a little higher, at least in some of the conden- 
sers) showed a slightly less loss. At 47° C. the loss had 
decreased to 8*0 per cent., and at 49°'5 C. to 6*5 per cent., 
only two-thirds its maximum value. The condensers were not 
all at the same temperature, and the indicated temperatures are 
therefore not exact. But they show unmistakably a maximum 
value of the condenser loss, or energy converted into heat, at 
about 39° C, and beyond that a very considerable diminution. 
No further readings were taken until the condensers had 
risen several degrees, when it was suddenly noticed that one 
pair was hotter than the others and getting soft. The ther- 
mometer in a cooler pair registered 59°, but the warmest pair 
was considerably higher. The loss was astonishingly large, 
but the condenser had not broken down. Moreover, the 
" leakage current " had not greatly increased, for while 839 
volts gave 1'50 amperes at 49°' 5, 433 volts gave '80 ampere 
at 59° C. To be in exact proportion to the voltage the current 
should have been *774 instead of *80 at the higher tempera- 
ture, a comparatively small difference. 

To find so large a loss in commercial condensers of good 

40 X)r. E. H. Cook on Experiments 

repute was a surprise to us. To find a well marked maximum 
as the temperature rose, beyond which the loss decreased as 
the beeswax and rosin composition softened was a second 
surprise. To find so large a loss as the last observation 
shows without the condensers giving way, and without any 
very large leakage current, was a third surprise. 

In order therefore to verify these results by a totally 
different method, and to determine as accurately as possible 
the losses in some paraffin-paper condensers which we possessed 
which showed relatively very small heating effects, we built 
a special form of calorimeter, into which the condensers could 
be placed and the heat directly measured. The calorimeter 
was copied after the large respiration calorimeter which one 
of us designed for experiments under the patronage of the 
U. S. Government, and which is located at Wesleyan 
University. The description of the calorimeter and the 
results obtained with it are reserved for a subsequent com- 
munication. We will only add that they fully confirm the 
unexpected results obtained by the resonance method given 
above concerning the dissipation of energy in beeswax and 
rosin condensers. 

Wesleyan University, 

Middletown, Conn., July 1, 1898. 

III. Experiments with the Brush Discharge. 
By E. H. Cook, D.Sc. (Bond.), Clifton Laboratory, Bristol* . 

[Plate I.] 

rt^HE ordinary phenomena which accompany the brush- 
JL discharge are well-known, but in view of the recent 
extension of our knowledge of electric discharges in high 
vacua, it seemed desirable to study the subject a little more 
closely. The following experiments have been made with 
this object. 

Most of the results have been obtained with an ordinary 
Wimshurst machine with 15-inch plates, but they have also 
been produced with the discharge from an induction-coil, as 
well as, though less readily, with a plate frictional machine. 

In experiments requiring the production of the brush for 
a short period the machine was turned by hand, but where a 
long-continued effect was desired, motion was obtained by 
the use of one of Henrici's hot-air motors. The number of 
revolutions of the plates was counted by means of a tacho- 
meter, and the number of volts was taken as being about equal 

* Communicated by the Author, having been read before the British 
Association at Bristol, 1898. 

with the Bru*h Discharge. 41 

to what would be produced by the same speed of rotation 
between knobs of one centimetre diameter. (See Joubert, 
Foster and Atkinson, ' Electricity and Magnetism/ p. 103.) 

In all cases the results have been produced at ordinary 
atmospheric temperatures and pressures, but, of course, the 
brilliancy of the effects varies with the climatic conditions. 
For this reason no attempts have been made to measure the 
size of the brush, because it differs so much from day to day. 
The experiments described can be reproduced under varying 
conditions, and the effects may therefore be regarded as normal 
accompaniments of this kind of discharge. 

1. Shape of the Brush-Discharge. 

As is well known, if a discharge of negative electricity takes 
place from a pointed conductor, and if the point be examined 
in a darkened room, it will be seen that it is surrounded by a 
faint spot of light of a violet, or violet-blue colour. If, on the 
contrary, a similar experiment be made with positive electricity, 
the point will be seen to be surrounded by an innumerable 
number of lines of light of a similar violet-blue colour, forming 
what is called the brush. It is stated, on the authority of 
Faraday, that the glow which surrounds the negative point is 
separated from it by a dark space. Undoubtedly this is the 
case when the discharge is taken in rarefied air, but at the 
ordinary pressures I have been unable to detect it, although 
the brush has been examined under the microscope. The 
glow seems to be in contact with the point. The positive dis- 
charge, however, behaves differently. When carefully observed 
it is seen that the lines do not start from the exact end but at 
some slight distance away (2 or 3 millims.). They appear to 
keep in a bunch for a little distance and then to diverge. The 
size of the positive brush is much increased by the proximity 
of an earth-connected plate or sphere, and the outline of the 
luminous portion is altered by the shape and nearness of this 
" earth." Thus, when an u earth " is some distance away, the 
emanation from a point may take the shape of a fan with the 
side lines at right angles to the point as in the figure. If to 
such a brush an " earth " be brought to within a few centi- 
metres, the lines will curve themselves round and the angle 
of the fan instead of being 180 degrees will become much 
less. The glow at a point giving negative electricity becomes 
brighter if an earth-connected body be brought near, but it 
does not alter in size until the body is very close (less than a 
centimetre), when small sparks pass between the point and 
the body. 


Dr. E. H. Cook on Experiments 

The angle of the point makes a considerable difference to 
the shape of the positive fan. If the end consists of a small 
angle, for example a needle, the bounding lines of the fan 
enclose a small angle, and the whole of the luminous portion 

45 000 




26 000 






is very small. As the angle increases so does also the angle 
of the fan. The figures show the kinds of discharge obtained 
from brass points of varying angles. They were obtained 
from the positive side of a machine, and under similar con- 
ditions of proximity to earth-connected plates. The dis- 
charges are drawn of about actual size, but the wires and 
points are drawn larger than they were for the sake of 
clearness. The approximate differences of potential between 
the knobs of the machine when giving these discharges are 

ivith the Brush Discharge. 43 

stated. The appearance of the discharge obtained from 
concave ends is also shown. Concave terminals behave like 
angular ones. 

2. Force of Wind from Points. 

The mechanical force exerted by the strongly electrified air- 
particles which are repelled from the points is well known, 
and the experiment of blowing out a candle is one of the 
commonest shown as illustrating the action of points. The 
magnitude of the force was roughly measured by causing 
the discharge to play upon one pan of a delicate Robervahl 
balance and measuring the weight necessary to restore the 
equilibrium. Care was taken to use the same point and to 
change the polarity by reversing the machine. This was the 
only way in which comparative experiments were possible. 
If the attempt be made to measure from two different points 
it will be found that the minute differences in the points, 
notwithstanding every care to make them precisely similar, 
will show themselves by entirely altering the appearance of 
the brush, and great differences in magnitude will be observed, 
even with the same kind of electricity, from the different 

In order that vibration might be avoided as far as possible, 
the measurements were made on a stone slab, built up from 
the foundations of the laboratory. The following results 
were obtained : — 

Wimshurst machine 15 in. plate. 

No. of sectors on each plate . . 16. 
Size of sectors 8 sq. cm. 

(1) Speed of revolution 450 per minute. 

Potential-difference corresponding to 

this speed, about 43,000 volts. 

Force from positive brush equal to 

a weight of 0*29 gramme. 
Force from negative brush equal to 

a weight of 0*24 gramme. 

(2) Speed of revolution 182 per minute. 

Potential-difference 35,000 volts. 

Force from positive brush equal to a 

weight of 0-08 gramme. 
Force from negative brush equal to 

a weight of 0*066 gramme. 
The best distance of brush from pan 

of balance 0*04 metre. 

44 Dr. E. H. Cook on Experiments 

Induction- Coil : — 

Length of primary wire . 200 feet (500 turns) . 
Length of secondary wire . 60,000 feet (490,000 turns). 
Capacity of condenser . . 2 microfarads. 
Length of spark in air . . 0*07 metre. 
Corresponding to 
* Potential-difference of about . 63,000 volts. 
Force from positive brush equal to a weight of 

0*01 gramme. 
Force from negative brush equal to a weight 
of 0'01 gramme. 

It will thus be seen that the magnitude of the force is 
greater from the positive side of the machine than from the 

An attempt was now made to find the maximum distance at 
which this mechanical disturbance could make itself felt. 
For this purpose the following experiment was arranged. A 
single fibre of unspun silk was stretched across the field of 
view of the microscope, and on this was hung a little paper 
index which was blown aside by the wind from the point. 
The maximum distance at which any deflexion could be 
observed was then noted, and in this way a comparison was 
instituted. Care was again taken not to use different points, 
but to alter the polarity by reversing the machine. Also, 
during the experiment, the whole apparatus was carefully 
protected from extraneous currents of air. It was found that 
with the machine with a potential-difference of 33,000 volts 
the positive brush produced an effect at a distance of 0'6 metre, 
the negative at a distance of 0*48 metre. When the potential 
difference had fallen to about 25,000 volts the distances 
observed were for the positive 0*32 metre, and for the negative 
028 metre. With the coil giving a spark of 4 centimetres 
(41,500 volts) the positive brush affected the thread at a 
distance of 32 metre, and the negative moved it at the 
same distance. 

These results therefore confirm the former ; for the positive 
brush affected the silk at the greater distance, and also pro- 
duced the greater pressure upon the pan of the balance. But 
they show how very quickly a moving electrified particle of 
air is brought to rest by surrounding air. 

* These details are given in order that an idea may be formed of the 
kind of apparatus worked with, and that if any one should desire to 
repeat the experiments he may know what results to expect. The 
actual numbers will vary with the apparatus. 

with the Brush Discharge. 45 

3. Electrical Action at a Distance. 

If an electroscope or a leyden-jar be placed at some distance 
from a point from which a brush-discharge is taking place, 
it will become charged. If the brush be a positive one the 
electroscope or jar will be charged positively ; if the brush be 
a negative one it will be charged negatively. 

The distances at which the effects make themselves evideut 
vary with the potential and the atmospheric conditions. 
Statements of lengths therefore are only valuable as allowing 
of comparison being made between experiments which are 
performed at the same time and in the game laboratorj^. The 
same relation will, of course, hold, but the actual measure- 
ments will be different. With these reservations the following 
are given : — 

With the plates of the machine revolving at the rate of 
450 revolutions, and giving a difference of potential of about 
43,000 volts, an electroscope was affected at a distance of 
1'8 metres from the point. No difference was made by altering 
the polarity. 

When the plates were revolving at the rate of 105 revolu- 
tions, and the potential-difference was 25,000 volts, the 
same electroscope was affected at a distance of 1/0 metre. 

The size of the collecting-plate of the electroscope makes a 
considerable difference in the ability of the instrument to 
become charged. The larger this plate the greater the 
distance at which it can be charged. A point on the plate 
reduces the distance somewhat, probably because it allows 
the electricity to escape as fast as it can be collected. 

No increased effect could be obtained by increasing the 

The shape of the point which gives the best results is 
acute. This is different from the effect in producing lumi- 
nosity. In that case it will be seen that a larger luminous 
brush is obtained with a greater angle up to 90°. 

Notwithstanding the enormously greater difference of 
potential produced, the brushes from the coil used were quite 
unable to influence the electroscope at distances equal to those 
obtained from the machine. Thus, when the coil was giving 
sparks of 0*07 metre in length and thus causing a potential- 
difference of about 63,000 volts, the positive brush was only 
capable of charging the electroscope at a distance of 0*62 metre 
and the negative at the same distance. 

The same relative results were obtained when the brushes 
were used for charging a leyden-jar. Thus, with a difference 
of potential of about 40,000 volts, a jar was charged by the 

46 Dr. E. H. Cook on Experiments 

positive brush from the machine at a distance of 0*55 metre. 
The negative brush charged it negatively at the same distance. 
With a concave terminal this distance was reduced to 
0*50 metre. 

With the coil giving a spark of about 0*7 metre (63,000 
volts) the jar was charged at a distance of only 0*14 metre. 

But, as has been already stated, these distances vary greatly. 
With favourable atmospheric conditions, and a plate on the 
electroscope 16 by 23 centims., it was found that the electro- 
scope could be readily affected at a distance of 4 metres from 
the point attached to the 15-inch Wimshurst machine. On 
comparing these distances with those obtained when measuring 
the mechanical force of the wind from the point, it is seen 
that the electrical effects are felt at a much greater distance, 
amounting, if we take the maximum distances, to more than 
six times. 

It is not at all necessary that the conductor ending in the 
point should be directed towards the electroscope. The 
instrument will be found to be influenced in almost any 
position ; in fact, when experimenting in frosty weather, I 
have obtained results at equal distances all round the point, 
even in the exactly opposite direction. It is therefore clear 
that the point is the centre of a disturbance which radiates 
from it in all directions. 

The interposition of objects between the point and the 
electroscope gives interesting results. A wire- cage, if com- 
pletely covering the instrument and earth-connected, will 
prevent it from being charged, but if it is not completely 
covered, even if only one side of the base is tilted up, the 
instrument will become charged. The same wire-cage held 
between the point and electroscope has no effect. Plates of 
metal, wood, or other material placed between, do not alter 
the effect unless they are either close up to the point or to the 
electroscope. In both cases a diminished effect is observed. 

Experiments were tried with the view of discovering if the 
effects could be produced through substances ; i. e. if like the 
Rontgen rays the brush-discharges possessed any penetrative 
power. Although by this mode of experimenting definite 
proof could not be obtained, several of the results are instruc- 
tive. Thus, if the electroscope be placed at from 2 to 3 
metres away from thepoint, and a board 56 by 78 by 15 centims. 
be interposed midway in the path of the discharge, the leaves 
are easily affected, almost as easily as if the board were absent. 
The same thing happens if sheets of metals are interposed. 
The direction of the point does not affect the result. If the 
sheet be placed within 30 centims. of the point, the effect on 

ivith the Brush Discharge. 47 

the leaves is not produced. If the sheet be placed near to 
the electroscope {within 10 centims.) no effect is produced on the 
leaves, but as the distance is increased the leaves diverge more 
and more ; thus at 50 centims. the effect is nearly as much as 
when the board is midway. It is impossible while witnessing 
these experiments to avoid calling to mind the similarity of 
the effects produced to those which one sees on the coast when 
a long billow rushing onwards towards the shore meets with 
a solitary rock in its path. The rock is grasped on all sides, 
but while immediately behind it the water is comparatively 
still, the waves soon curl as it were round the sides and meet 
each other at a short distance behind. The result being that 
the effect of the rock is to interfere with the wave for only a 
short distance immediately behind it. Beyond that there is 
as much commotion in the line from the middle of the rock to 
the shore as at the edges. 

If while the electroscope leaves are diverged and the 
machine working, a person walks between the point and the 
instrument, the leaves will sway with his movements, but will 
not fall together. 

The production of the effect all round the point extends to 
positions under the machine, for I have repeatedly obtained 
the divergence of the leaves when the electroscope was placed 
directly under the machine, which stood on an inch board on 
the bench, which had a two-inch top. 

4. Chemical Action produced by the Brush-Discharge. 

The formation of ozone by the working of an ordinary 
electrical machine is well known, but whether positive or 
negative is more active in its production has not been 
investigated. It was therefore determined to examine the 
action of the brush-discharge in producing this and other 
chemical changes. The whole of the work contemplated has 
not been completed, but some results of interest may be 
mentioned. Considerable difficulty was experienced, in con- 
sequence of the high potentials used, in preventing leakage. 
In some cases leakage took the form of brushes at other 
places than the points required. In such cases the results 
were useless. In consequence of this, whenever it was 
necessary to lead the brush away from the point only the 
most strongly insulated wire was used, and this was first 
specially examined in order to see if any cracks or thin 
places were in it. Whenever the wire had to be led into 
flasks or bottles it was found that the best material for bungs 
was solid paraffin. But it was found very difficult to produce 
good brushes inside glass vessels, because the interior surface 

48 Dr. E. H. Cook on Experiments 

very soon became rapidly charged, and this seemed to pre- 
vent the production of the brush. 

Experiments with Potassium Iodide. 

The chemical action produced by the brushes in air may be 
the formation of ozone, the production of oxides of nitrogen, 
and perhaps other less known combinations. Potassic iodide 
would be decomposed by ozone and the oxides of nitrogen, 
and it is therefore a suitable substance to experiment with. 
After many trials the simplest apparatus was found to be the 
most useful. The points attached to the positive and negative 
sides of the machine were placed at such a distance above the 
surface of some standard potassic iodide solution that no 
actual spark from the point could pass to the solution (with 
the machine and potential used, this was anything beyond 
2 centim.). This precaution was taken in order that no breaking 
up of the iodide should take place in consequence of the 
spark passing through it. The solution was contained in glass 
or porcelain dishes, and as solution of potassic iodide (unless 
perfectly pure) becomes slightly coloured when exposed for 
some time to air and light, a similar quantity of the standard 
solution w 7 as placed in an extra dish of equal size at the same 
time. At the conclusion of the experiment the amount of 
iodine set free from this blank was estimated and subtracted 
from that produced under the brushes. 

The results obtained have differed among themselves as far 
as actual amounts of iodine set free at one brush compared 
to the amount set free at the other, but they have all agreed 
in this important particular, that the amount of iodine produced 
by the negative brush from the machine is always very much 
greater than the amount produced by the positive in the 
same time. 

With points prepared as similarly as possible, and every 
precaution taken to avoid leakage, it was found that from, five 
to eight times as much iodine was set free by the negative 
brush as by the positive. The following are details of a 
typical experiment : — 

Speed of rotation of plates of machine, 300 per minute. 

Potential-difference about 40,000 volts. 

Distance of points from surface of solution, 4 centims. 

Amount of iodine liberated by the negative brush over that 
in blank in half-hour, 0*000762 gramme. 

Amount of iodine liberated by the positive brush over that 
in blank in half-hour, 0*000 127 gramme. 

The greatest amount of iodine liberated in one hour in this 
way amounted to 0'001778 gramme ; assuming this to be 

with the Brush Discharge. 49 

produced by the formation of ozone, which then decomposes 
the iodide, it would correspond to the formation of 
0'000112 gramme of ozone. 

Condensing the electricity produces very little effect. For 
with the jars on, similar results have been obtained to those 
without the jars. Thus with points 4 centims. away and a 
potential-difference of about 40,000 volts, 0*001778 gramme 
of iodine was produced at the negative brush, and 0*000190 
gramme at the positive in one hour. 

A diminution of potential-difference reduces the amount of 
chemical action ; but the substance of which the point is 
composed is immaterial. Thus, with 25,000 volts difference, 
0*000698 gramme of iodine was set free by the negative in 
half an hour, as against 0*000762 gramme with 40,000 in the 
same time. 

The distance of the point from the surface of the liquid 
makes an important difference. It has already been stated 
that all the experiments were made when the points were 
farther away than that at which a spark could pass. The best 
effects are caused when the points are just farther away than 
this. Thus when the points were 12 centims. away 0*000021 
gramme was liberated by the negative, as against 0*000063 
gramme when 8 centims. away, and 0*000508 gramme when 
4 centims. away. 

Similar experiments to the above made with the brushes 
obtained with the coil have given different results. In this 
case it was found that invariably the brush from the positive 
terminal liberates the greater amount of iodine. The difference 
between the amounts produced by the positive and negative 
brushes is not, however, so great as that by the machine. In 
this case also we find differences between individual experi- 
ments, but as before always an excess by the same pole. 
The average of many experiments gave from three to five 
times as much iodine set free under the positive brush to 
that set free under the negative. These numerical results 
apply to the coil whose dimensions are given above. 

It was found that with the machine and coil as described, 
a much greater quantity of decomposition was caused by 
brushes from the machine than from the coil. This corre- 
sponds to the amount of visible brush produced. 

Actum on other Substances. 

The power of the brushes to produce other chemical 
actions has been partially investigated. So far no reducing 
actions have been observed, but oxidizing ones are always 
present. In these cases with the machine brushes more action 

Phil. Mag. S. 5. Vol. 47. No. 284. Jan. 1899. E 

50 Dr. E. H. Cook on Experiments 

takes place at the negative than at the positive, and generally 
the same conditions apply as in the case of potassic iodide. 
Thus about four times as much iron is oxidized from the 
ferrous to the ferric condition by the negative as by the 
positive brush. 

5. Effect on the Electrodes. 

It is stated* on the authority of Wheatstone that : — 
" Metallic dust is in every case torn away from the electrode 
by the brush discharge/' This statement is one which would 
be supposed to be true when it is considered that such is 
known to be the case with the spark-discharge. But it is 
contradicted by spectroscopic evidence, for the spectrum of 
the glow-discharge shows no trace of metallic lines. It is the 
same u whatever the nature of the metal, and is due solely to 
the incandescent gas/'f In order to test the statement 
several experiments were made. The brushes were obtained 
from copper points, and made to play upon the surface of 
some dilute nitric acid placed in dishes under them. The 
points were brought as near as possible to the surface, i. e. 
as near as possible without producing a spark-discharge. 
They were so near that minute waves were formed during 
the whole time of the experiment by the wind from the points. 
No possible loss of copper could therefore occur if any were 
torn off from the point. The machine was run at its highest 
speed, and thus the greatest difference of potential available 
(from 40,000 to 50,000 volts) was obtained. The experiment 
was continued for two hours, during the whole of which time 
very fine luminous brushes were being produced at each point. 
At the conclusion of the experiment the acid was carefully 
concentrated by evaporation and tested for copper. Not a 
trace could be found in either. Thus showing that no metallic 
particles icere torn off the points. Moreover, I have examined 
microscopically a pair of points which I had cut and prepared 
from a piece of guttapercba-covered wire. These points have 
been continuously in use for some months, and it is certain 
that at a low computation brushes must have been drawn 
from them for at least 150 hours. The edges seem as sharp 
as when cut, and sensible alteration has not taken place in 

Of course these results apply to the potential and quantity 
worked with. Higher potential and increased quantity may 
give quite different results. 

* Silvaims Thompson's l Elementary Lessons/ p. 305. 
t .Toubert, Foster and Atkinson, ! Electricity/ p. 103. 

with the Brush Discharge . 51 

6. Action of the Brush-Discharge upon Photographic Plates, 

The productions of chemical actions by the brush-dis- 
charges immediately gave one the desire to try the action 
upon photographic plates. The first experiments of this kind 
were made shortly after the discovery by Rontgen ; but as 
they did not lead to any results which were of such general 
interest as the latter, the study was discontinued for a time 
but taken up again recently. 

In these experiments, as in the former, the brush from the 
machine gives better results than that from the coil, but the 
positive seems to be more effective than the negative. In 
the latter case a longer exposure is necessary, and sometimes 
less definite results are obtained. Films of different kinds 
were used, but because of the greater ease in development, 
ordinary " slow " plates were most frequently employed 
("Ilford Ordinary"). 

Action of Brush-Discharge upon a Sensitive Film. 

If a photographic plate be placed on the table in a dark 
room with the uncovered film upwards, while the positive or 
negative brush from a machine or coil be arranged at some 
distance (say 4 feet) above it and the point turned towards it ; 
the plate after development will be found to be " fogged," 
showing that a decomposition of the silver salts has been 
brought about, similar to that produced by exposure to light. 
With the brush from the positive terminal the reduction of the 
silver compound is fairly uniform all over the plate, but when 
the negative is used there are " blotches " in several places, 
showing that the reducing action has taken place in some 
spots more than in others. There is therefore produced at 
the point an emanation of some kind, whether it be an undu- 
latory movement or a stream of particles, which possesses the 
power of reducing silver salts. 

If between the point and the sensitive film a solid object 
be placed, a shadow of the object will be thrown upon the 
plate. This shadow is sharp if the object is close to the 
plate, but its edges are ill-defined if it be at some distance 
from it, in this particular exactly resembling light. PI. I. 
(figs. 1 and 2) shows this, the object being a piece of sheet- 
zinc cut into the shape of the letter H and placed firstly at 
1*5 centim. away from the plate, and secondly at 10 centims.^the 
point being about 30 centims. away and the potential about 
35,000 volts. By measuring the size of the object and images, 
and the distance between them, it is easy to find the position 
of the source from which the rays emanate. When this is 
done it is found that the rays proceed from the extreme end 

E 2 

52 Dr. E. H. Cook on Experiments 

of the point and not at some distance from it. The visible 
divergence of the brush does not start exactly from the point 
but at some little distance in front of it. It would seem 
therefore that the point of the emanation of these rays does 
not coincide with that of the visible part of the brush-dis- 

The next step was to find if the effect could be produced 
after reflexion. This was shown with the following apparatus. 
A cardboard box was made of the shape shown in sketch. 


A B is a mirror, the positive terminal of the machine, D an 
object, and E the photographic plate. Internal reflexion was 
carefully prevented and the experiment made in the dark 
room. The point was about 20 centims. from the mirror and the 
mirror about the same distance from the plate. The object, 
which was a piece of sheet-zinc, was from 1 to 5 centims. from 
the plate. When the machine was worked, and no mirror in 
position, it was found that the exposed plate was unaffected, 
the rays we may imagine being absorbed by the sides of the 
box. This experiment also shows that the fogging of the 
plate is not caused by any ozone which may accompany the 
discharge. When, however, the mirror is in position a dis- 
tinct image is formed upon the plate after a slight exposure 
(5 minutes was usually given). Supposing the effects to be 
caused by the emission of electrified particles from the terminal 
it might be imagined that these would be reflected from the 
mirror and so turned from their course as to impinge upon 
the plate. In order to discover if this explanation be correct, 
the mirror A B (which was of glass in the first experiment) 
was replaced by a sheet of metal. Xow, if electrified particles 
fall upon this and it be earth-connected, their electricity will 

with the Brush Discharge, 53 

be immediately discharged, and therefore when they subse- 
quently impinged upon the plate no effect would be produced. 
This was found not lo be the case, as is shown by PI. I. (fig. 3), 
which was produced by employing a piece of tin for the 
reflecting surface instead of glass. 

That the metal plate did receive the impact of electrified 
particles was shown by the fact that sparks could be drawn 
from it from time to time. 

In view of the statement of Tesla, that the efficiency of 
the metals as regards their reflecting powers for #-rays 
follows their order in the voltaic series, an attempt was made 
to test if any variation of reflecting power could be detected 
in the present case. For this purpose the metal reflecting- 
plate was varied, but the experimental results show that no 
conclusions can be drawn indicating any connexion between 
the reflecting-power and position in the electrochemical series. 
In fact the reflexion seems to depend upon the brightness of 
the surface, and therefore agrees exactly with the reflexion 
of light. 

7. Penetrative Effects produced by the Brush-Discharge. 

The remarkable results obtained by Hontgen and others 
induced an attempt to imitate the effects by the brush-discharge. 
For this purpose a sensitive plate was wrapped in brown 
paper (two folds), and on the paper were placed sundry small 
articles, such as coins, keys, &c, and the whole exposed to the 
brush-discharge. The experiment was made in a darkened 
room, and the point placed at about 5 to 6 centims. above the 
coins, the plate lying on the table. An exposure of 30 minutes 
was given, and on developing the plate the outline of the 
articles was distinctly shown. This experiment was repeated 
with brushes of different polarity and source {i. e. coil and 
machine), and in every case the same results were obtained. 
Probably the cause of this action is that the substances 
become charged and act inductively upon the silver salt in the 
sensitive film, causing a partial decomposition or production 
of a " latent image/' which decomposition is carried still 
further in the process of development of the plate. The next 
step was to see if the outline of the bones could be produced 
without the flesh. Numerous experiments were made, but 
although in all cases the outline of the hand could be repro- 
duced in no case did the bony skeleton show itself. 

None of these effects could be obtained when a piece of 
vulcanized fibre was included in the wrapping. This substance 
has been shown by Giffard and others to be impervious to 
#-rays, and, of course, it is equally impervious to light. 

54 Dr.E.H.Cook on Experiments 

As the thickness of the wrapping increased, definition on 
the developed plate became less and less. Thinking the effect 
might be due to the fact that the brown paper wrapping was 
not light-tight experiments were made to test this idea, and 
it should here be mentioned that the brown paper referred to 
is that which is used, by photographic-plate makers to wrap 
sensitive plates in. Firstly, a plate wrapped in one fold was 
exposed for one hour to the light of an ordinary 8 c. p. glow- 
lamp illuminated by a 105 volt alternating current. A negative 
was obtained clearly showing images of the objects placed on 
the brown paper. This seemed to support the idea. Then 
a wrapped-up plate was simply exposed to diffused daylight 
for two hours. A very faint and blurred image of the coins 
&c. was obtained. Thus showing that the paper was not 
absolutely light-tight. Moreover, by exposing to very power- 
ful light, such as that from burning magnesium and the 
lime-light, clearer effects were obtained, showing that the 
more powerful light is capable of getting through the paper 

8. Reproduction of Prints fyc. by Brush-Discharge. 

Whilst engaged in obtaining a perfectly light-tight 
wrapping for the plates it happened that a piece of ordinary 
notepaper was used and the whole exposed to the action of 
the brush. On developing the plate a clearly-defined image of 
the watermark of the paper was produced (fig. 4) . This induced 
trying to copy in a similar way printing, writing, pictures, 
&c. In every case this has been done with complete success. 
If a photograph, or a drawing, or printing, or writing be placed 
in contact with the sensitive film and exposed to the brush- 
discharge, a clearly-defined and very sharp reproduction is 
obtained. This seems to be more readily produced by the 
positive brush than by the negative. The effect on the plate 
can also be produced if the drawing be not in contact with 
the film but separated from it by one or two layers of paper 
or cardboard. In the latter case, however, the definition is 
not so good. 

The effects produced when the drawings are in contact 
cannot be produced simply by keeping the plate and print in 
contact, at any rate for the same time as was used in my 
experiments, but can be brought about by exposing the plate 
to a powerful light, providing the wrappings are not too 

One of the results of these experiments, of a somewhat 
startling character perhaps, is that the writing on a letter 
inside an envelope can be reproduced. It may be some 

loith the Brush Discharge. 55 

comfort, however, to know that there is considerable difficulty 
in recognizing the words owing to the folding of the paper, 
and thus one word coming immediately on the top of another. 
Fig. 5 shows the result obtained when a printed invitation- 
card enclosed in an envelope was exposed to the brush- 
discharge, the envelope being placed on the sensitive film. 
Notice the texture of the paper, and the opacity caused by 
the gum and double thickness of paper. 

Light from the Discharge. 

The great similarity between the effects recorded in the 
foregoing pages and those produced by ordinary light led to 
experiments being made to compare them, 

The first point which suggested itself was to see whether the 
interposition of a body which was transparent to ordinary 
light, between the point and plate, made any difference in the 
result. A plate of clear glass was used, and the shadow of 
an object obtained. No difference was observed in the sharp- 
ness of the image. The glass was now blackened with lamp- 
black until it was so opaque to light that the flame of an 
ordinary candle could not be seen through it when it was 
held at a distance of 5 centims. from the flame. When this 
was held between the point and the photographic plate no 
effect whatever could be obtained. 

Again, the law of inverse squares was proved in the follow- 
ing manner : — A small cross cut out of thin sheet-zinc was 
placed at a certain distance from the sensitive plate and its 
shadow obtained. The plate was now moved so that the 
shadow should fall upon a different part of the film, and the 
object was placed at a different distance away. Another 
shadow was now obtained. This was of a different size to 
the former. The two figures were then measured, and the 
sizes compared with the object and the distance of the brush. 

These results therefore indicate that the effects are produced 
by the light which the brush emits. Moreover, after many 
trials, I have been enabled to reproduce all the effects with 
artificial sources of light. Many of them can be produced by 
employing daylight, and probably all could thus be formed, 
but the length of exposure required has hitherto prevented 
this from being done. 

But, notwithstanding this apparently simple explanation, it 
is quite possible that we have something else taking place at 
the same time. 

Suppose that the point is the centre of a disturbance from 
which waves are emitted. These waves will be of various 
lengths, some capable of affecting the optic nerve, and some 

56 On Experiments with the Brush Discharge. 

of shorter and some of longer wave-length than these. The 
reduction of the silver salts in the sensitive film may be 
caused by more than one kind of these waves. Before then 
it is possible to say that the effects are caused by the actinic 
power of the waves of short wave-length only, it is necessary 
to separate these waves in some such manner as that employed 
by Tyndall to separate heat-waves from those of light. 
Various experiments have been made to do this, but hitherto 
without success. The investigation is being pursued in this 

The actual amount of light given out by the brush- discharge 
is not much, and in the reflexion experiments which have 
been described it was found impossible for the most delicate 
eye, even after being kept in absolute darkness for a con- 
siderable time, to distinguish the outline of the object; but the 
shadow was nevertheless easily produced on the photographic 
plate. The numerous experiments which have been made 
show that, so far as one can judge, the effects produced by 
the brush are far more definite than would be expected, when 
the very small luminosity of the discharge is remembered. 
Or, in other words, that the emanation from the point contains 
a much larger proportion of rays capable of bringing about 
chemical decomposition than would be supposed when we 
remember only its luminosity. 

In order that some approximation might be obtained between 
the light-giving power of the discharge and ordinary light, the 
following experiment was made. A comparison was first ob- 
tained between the light from a standard candle and the smallest 
burner procurable, the gas being burnt under the usual con- 
ditions for regulating the pressure. This burner was now 
compared with the light from the brush. The ordinary Bunsen- 
photometer, as used in gas-testing, was employed, and the 
light from the machine carefully screened off. In order that 
a good brush might be obtained an u earth " was placed near 
the terminal. The actual figures obtained varied with the 
climatic conditions, but not so much as would be expected, 
and the following numbers may be taken as an approximation, 
but, of course, only an approximation, to the relative lumi- 
nosity of a standard candle and the brush from the machine 

Distance of candle from photometer ... 60 inches. 

Distance of light „ „ ... 3' 2 5 inches. 

Therefore, Candle : Light : : 3600 : 10*56. 

Distance of light from photometer . . .56 inches. 

Distance of positive brush from photometer . 4 „ 
Therefore, Light from positive brush : Candle : : 1: 267,200. 

Irregularities of Meteorological and Magnetic Phenomena. 57 

The light from the negative point is less than that from the 
positive. Brass points were used when making this com- 
parison, but no difference was observed with other points. 
The potential-difference was about 30,000 volts. 

Comparison of the Actinic Poiver of the Brush with that 
of Light. 

In order to roughly test this an ordinary negative was 
taken and fixed in a frame together with a piece of bromide- 
paper. The frame was then exposed in the dark room to the 
light from a standard wax-candle. The same negative was 
then treated in the same way and exposed to the brush-dis- 
charge for a given time. The prints were then fixed and 
compared in order to see if they were over or under exposed. 
The experiments were then repeated until the effects were 
equal. After many experiments the nearest comparison that 
could be obtained was that the light from the candle at a 
distance of 15 centim. and 20 seconds' exposure produced 
the same effect as that from the positive brush at the same 
distance and 15 minutes' exposure. From this the relative 
powers would be as 1 to 45. The potential-difference when 
making these tests was about 30,000 volts. 

On comparing this number with that obtained from the 
photometric experiments it will be. seen how widely they differ, 
and it may therefore be considered as certain that the emana- 
tion from the point consists very largely of those waves 
which are capable of bringing about chemical changes. 

IV. On the Analogy of some Irregularities in the Yearly 
Range of Meteorological and Magnetic Phenomena. By 

[Plate II.] 

AT the Toronto Meeting of the British Association f I 
called the attention of Section A to the fact that if the 
normal temperatures for every day of the year are plotted 
down in a curve, such curves are strikingly similar for 
stations spread over a very large area. An area which is 
larger than our continent for some of the particulars shown 
by these curves, while for others it extends over Western 
Europe only, or over part of it. 

I am now able, to a certain degree, to give an answer to 
the query at the end of that paper : " Is it temperature alone 
of which the irregularities are so extremely regular ? How 

* Communicated by the Author. 

t Paper published in the Phil. Mag. for May 1898. 

58 Dr. van Rijckevorsel on Irregularities in the Yearly 

does the barometer behave ? Do the winds, do the magnetic 
elements, show something pointing to a common origin ? " 
The answer is most certainly a positive one, as I hope to 
show now. 

On the diagram (Plate II.) the uppermost curve, marked T, 
shows the temperature of every day of the year for the Helder 
in the Netherlands. The next curve, marked H, shows the 
horizontal, and the following one Z the vertical component of 
the earth's magnetic force as shown by the registering instru- 
ments at Utrecht. The curve marked R the rainfall at the 
same station. The curve marked P shows the barometric 
pressure at Greenwich ; the last curve, D, the magnetic 
declination at the same station. 

T shows the mean of 50 years' observations, H of 33 years 
between 1857 and 1896, Z of 30 years in the same period. 
For the magnetic data it is not always possible to use all the 
years for which values are available ; for if in any case a 
serious break occurs in a series, which cannot be safely 
bridged over by fictitious values, or if a discontinuity occurs 
such as may be occasioned by a slight alteration in the instru- 
ment or in the adjustment of the scale, it is, as a rule, necessary 
to reject that whole year — this for more than one reason, but 
chiefly on account of the secular variation which is so irregular. 
It is unnecessary, however, to explain this now at any length. 

E, is the result of the 40 years 1856-95, P of the 18 years 
discussed by Mr. Glaisher *, viz. 1841-58, and lastly "D of 
the 30 years 1865-94, of which 1 owe the three years which 
have not been published in the publications of the Observatory 
to the courtesy of the Astronomer-Royal. 

Of course not all these series are of the same quality; some 
are decidedly not long enough ; but although a large amount 
of material is still being computed, I am at present only 
able to show T what there is in this diagram. The 19 years 
for air-pressure are decidedly not enough, and at the same 
time all those who are interested in terrestrial magnetism can 
form a judgment for themselves as to the intrinsic worth, of 
the data for the vertical component. But for my present aim 
the material is about sufficient. 

All the curves have been smoothed down in the same 
manner as explained in my first paper on this subject. For 
the magnetic curves, however, a preliminary operation was 
thought to be advisable, because the effect of the secular 
variation throughout the year would otherwise tilt the curves 
and render them less easily comparable with the others. This 
very simple operation was to take the difference between the 
* 9th and 10th Reports of the British Meteorological Society. 

Range of Meteorological and Magnetic Phenomena. 59 

mean for the ten first days of the year and that for the ten 
last days, and to interpolate this difference over the days of 
the year. The secular change has been taken into conside- 
ration in no other way; so that the figures from which these 
curves are drawn are simply the mean of the scale-readings 
for a certain number of years. Except for my purpose, they 
have therefore no value whatever. 

The scales on which these six curves are drawn are also 
purely arbitrary. For H and Z even the scale-divisions 
as published in the annals of the Koninhlijk Nederlandsch 
meteorologisch Instituut at Utrecht have not been reduced to 
any of the customary magnetic units. The factors by which 
the original figures have been multiplied in order to form the 
ordinates of these curves, ranging from 4 to J, were simply 
chosen so that the consecutive maxima and minima of the 
different curves should be, on an average, approximative^ of 
the same importance. In other words, in the vertical dimen- 
sions the curves have been compressed or expanded so as to 
render the phenomenon I wish to show as prominent as 

And I think it will be admitted that it is prominent. Of a 
few doubtful points I will speak later on. But if we except 
these for the present, I think that it is perfectly apparent that, 
however dissimilar the general behaviour of the individual 
curves may be, every single maximum and every minimum in 
one curve is, with an astonishing regularity, repeated in all 
the others. Far from it that the curves should be parallel. 
In many cases a maximum in one curve occurs earlier or later 
than in most of the others. In still more cases, what is a 
large bump or a deep valley in one curve may find its corre- 
spondent in a hardly perceptible movement in another curve. 
A few of these differences in behaviour shall be, later on, 
explained away in a satisfactory manner. Others may be due 
to insufficience, as yet, of the data at command. But even 
if a certain — decidedly small — number of instances should 
remain where one of the curves really has a secondary maximum 
and accompanying minimum not shown by the others, why 
should it not ? It may be a local anomaly; it may be that 
indeed for, say, the barometer an agent enters also upon the 
scene which has less grip, or none at all, upon the thermometer 
or the magnetometers. There are certainly not many such 
exceptions to the general phenomenon shown in the curves 
under discussion. Indeed, they are so few that I incline to 
think that we may predict even now that as soon as we shall 
be possessed of a sufficient number of sufficiently good obser- 
vations in every case it will indeed be found that to every 

60 Dr. van Rijekevorse] on Irregularities in the Yearly 

irregularity in one yearly curve, either for any magnetic or 
any meteorologic phenomenon, a corresponding irregularity 
will be apparent in all the other curves for the same region, 
if not alwavs. as will be shown, for the same station. But 
even if some doubtful point should remain. J. think proof is 
so abundant even now that, even if I should be found ulti- 
mately to have overrated it in some instances, this can no 
more invalidate the principle. 

Another very valuable test for my opinion that the ana- 
logies pointed out here cannot be accidental, is furnished by 
Ihis diagram for Paris *. The uppermost line shows the 
temperature as resulting from 130 years' observations. The 
lower one shows the magnetic declination, but this is the 
result of only 12 years of observation between 1785 and 1796 
bv Cassini. Xot verv good observations we should now call 
them : there are several breaks and changes of continuity 
which I had to bridge over as best 1 could. Moreover, the 
readings were not taken rigorously at the same hour of the 
day. 1 think this is more than sufficient, with such a short 
series, to account for a certain number of doubts and queries 
which I cannot explain away. But upon the whole 1 think 
it will be admitted that the concordance between the two curves 
is inst as satisfactory as in the large diagram. Therefore, at 
least also in this respect, things a century ago were what they 
are now. 

One possibility I must point out. It may be that one or 
more of these curves ought in reality to be inverted. This 
seems preposterous. Still, with so many maxima and minima 
following each other in rapid succession, it is not so. (I have 
numbered 17 maxima; but I am perfectly sure that a larger 
number than this are just as real as these, only they are so 
small as not to be detected yet, as long as we are obliged to 
smooth down the curves to such an extent.) Take a maximum 
and the following minimum, which slightly precedes the cor- 
responding feature of the other curves, and invert the curve : 
they will then simply be converted into a maximum and a 
preceding minimum slightly lagging behind a similar feature 
in the other curves. One of my magnetic curves is inverted 
— needless to say which, because a scale may read from right 
to left or from left to right. But also I have had serious 
doubts as to the advisability of inverting one or two others. 

You will see at once that in some cases my vertical lines f 

* This diagram is not reproduced, as the text is sufficiently clear 
without it. 

f Continuous verticnl lines connect the maxima, lines of long dots the 
minima belonging together. 

Range of Meteorological and Magnetic Phenomena. 61 

indicate a maximum where, in good faith, such a thing is not to 
be discovered. This is quite true. I call attention to the 
lines 6, 13, 14, 15. There is certainly no, or hardly an, 
apparent maximum in the temperature-line there. But if we 
look for a moment at this larger collection of temperature- 
curves*, of which it is not possible to give such a large diagram, 
it will at once be seen that for the three last of these numbers 
the curves for the whole of the United Kingdom show a small 
but decided maximum in all these cases. For 13 this also 
appears in some curves for southern stations, such as Mont- 
pellier. On the other hand, at 6 the maximum which the 
curve for the Helder does not show is most distinctly shown 
by the curves of the stations to the south and east of the 

There is, I think, a most valuable use of the method here 
explained. For one phenomenon any of the maxima may be 
less apparent than for another. It may even be not at all so, 
(It would seem as if magnetism were a more sensitive organ 
than meteorology.) But as soon as some of the curves show 
a certain maximum, there is some presumption that the others 
ought to do so, even if they do not. In other words, that 
there is at that particular moment an influence at work which 
would manifest itself by creating a maximum in those curves 
also, if only our series of observations were long enough or the 
methods of observation sensitive enough. If at some moment 
of the year the vertical intensity or the rainfall show a 
maximum or minimum, even an unimportant one, the tem- 
perature and declination must show it too — it may be smaller 
or larger, or somewhat larger or later, but it must be there. 
And should it decidedly not be there, depend upon it that here 
is something worth investigating. 

But again the point now under discussion gives us a 
valuable clue to the direction in which we must look for the 
origin of the anomalies of our curves ; and this may lead 
perhaps in some cases to a guess as to their causes. From 
what has just been said it is of course very probable that the 
maxima in 14 and 15 are due to the influence of some cause 
which has its seat to the north or northwest of the Netherlands, 
the one in 13 to one in the southwest, but that in to an 
influence coining from the southeast. 

I must here refer again to a very remarkable minimum 
occurring in some curves on or about the 1st of July, which 
has been mentioned in the Toronto paper f. The mass of 
material which has since come into my possession has shown 

* These diagrams also are not reproduced, 
t Phil. Mag\ May 1898, p. 405. 

62 Dr. van Rijekevorsel on Irregularities in the Yearly 

me that this is indeed more widely spread and more enigmatic 
than I at first supposed. However, I think I am able now 
to point out with a certain degree of probability where it 
originates. This is the first case in which I really could do 
more than show the direction in which this origin ought to be 




Greenwich . . 

Vlissingen .. 
TheHelder .. 


This second diagram shows for a certain number of stations 
the daily mean temperature for the mouth of July and parts 
of June and August. The order in which the curves are 
arranged is more or less the same as that in which the im- 
portance of this minimum decreases. At the same time — and 
this is important — the order is such that the first places are 
very near some point to the west of Scotland, and that the 
others gradually go further and further away from that centre. 

Onlv four of these show a decided minimum : the most 
pronounced one occurs at Arbroath in the east of Scotland, 
and the next at Valencia ; the two others are Rothesay near 
the west coast of Scotland and Thorshavn on the Faroe Islands. 

After these places the minimum assumes a less decided 
character ; it is no longer a valley, but only a less steep 
incline of the ascending curve. These stations form another 
circle round Scotland ; they are : Berufjord. on the east coast 

Range of Meteorological and Magnetic Phenomena. 63 

of Iceland*, Christiansand, on the west coast of Norway, 
Greenwich, and the stations in the Netherlands. 

Much importance must be attached to the fact that at 
Stykkisholm, on the west coast of Iceland, this minimum is 
not at all perceptible, and hardly, if at all, at Brest. And 
some importance also to the fact that it sets in first at the two 
Scotch stations, next at Valencia and the Faroer. 

If you consider these facts I think that you will own that 
here is an anomaly which must have its origin at the coast of 
the west of Scotland, and probably at no great distance. For 
if this origin were at a greater distance from the Scottish 
coast, it would be hardly conceivable that its influence should 
die out so rapidly on the two straight lines Thorshavn, 
Berufjord, Stykkisholm, and Scotland, England, Brest 

Thus far 1 believe my conjecture is backed by what con- 
stitutes a certain amount of proof. But the road, or rather 
roads, which this minimum subsequently follows through 
part of Europe is decidedly bewildering. Brussels shows it, 
Paris most decidedly; so do Lyons, Montpellier, Triest, and 
Klagonfurt; while, curiously enough, south and north of the 
line uniting these stations, both in the German and the 
Italian stations, I cannot find a trace of it. 

But in the North of Europe also there are decidedly some 
traces of it. Both Copenhagen and Haparanda show them, 
so do Baltischport and Kem on the western side of the 
White Sea. In this part of Europe the boundary beyond 
which this feature is not traceable is a line passing w r est of 
Konigsberg, Petersburg, and Archangel. Some traces of 
this minimum are also to be found in a group of stations of 
which Warsaw is the northernmost. 

There is another fact which may be connected with the 
strange distribution of this minimum. Nearly in the whole 
of Russia and the North of Europe the highest point of the 
whole curve has a strong tendency to occur in August, in 
the very beginning for the eastern stations with a single 
maximum, while those with a double summer maximum show 
the highest one about the middle of August. This may, 
for instance, be seen in the curves for Thorshavn, Berufjord, 
and Christiansand. In the west and south of Europe, on the 
contrary, the July maximum, on an average, is decidedly, 

* I owe these northern stations to the kindness of Dr. A. Paulsen, 
Director of the Meteorological Observatory at Copenhagen. I owe thanks 
to a great many more gentlemen for the kindness with which material, 
even manuscript, was placed at my disposal ; but their names will come 
more naturally when the time shall ultimately have come to publish the 
whole of this investigation. 

64 Dr. van Rijckevorsel on Irregularities in the Yearly 

although not much, higher than the August one. This is an 
indication of the southern origin of this July maximum. May 
not this be an explanation ? Lnte in June Europe is invaded 
from a point off the Scottish coast by a strong minimum, but 
a week or so later by a strong maximum from the south or 
southwest, let us say tropical Africa. Therefore it is probable 
that the fight, so to speak, of these two features, in the first 
place causes the distribution just spoken of, and that the 
curious particulars w r hich accompany it must find their expla- 
nation in local circumstances. 

A wild hypothesis ? I know it is. But 1 give it in order 
to show how vividly some meteorological problems are put 
before us by this method, and how a path is shown at the 
same time to their evolution. For can there be much doubt 
but that, as soon as we shall be possessed of more data, not 
about temperature only, more and more of these problems must 
make great progress towards a solution ? One curve for 
magnetic declination or for rainfall may throw a ray of light 
on an intricate point which at once solves it. 

Another argument to the same purpose. There will be 
seen, not without some effort perhaps, a very slight minimum 
between the 10th and 15th of May in the three upper curves 
(PI. II.) . There are other features of the same importance, of a 
greater one evem There need be no fear of my discussing them 
all, for the very plausible reason that I do not know anything 
about them. They may either prove interesting some day, 
or vanish completely, when meteorologists shall be able to 
discuss longer series of observations. But this one is worth a 
moment's notice, for it is really the mark of the so-much 
discussed u Ice-saints." You will notice a trace of it in the 
three upper curves, not in the two lower ones. (The rainfall 
is too doubtful to be quoted here.) Well, these two lower 
curves belong to Greenwich, and you will see that the hand- 
mark, if I may express myself thus, of these cold saints 
tends to vanish in the British Isles. The temperature-curve 
for Greenwich indeed shows it still, that for Valencia perhaps, 
but certainly in Scotland it is not visible. Here again the 
southern, or southeastern, origin of this phenomenon, which I 
hinted at at Toronto, is confirmed by other curves than those 
for the temperature. 

As hinted at before, some of these curves leave for the 
moment a doubt. It will be seen that between July and 
November I left in some of the vertical lines a break in the lower 
regions of my diagram. In nearly every case it is only the 
barometer that is at fault. I already have stated the cause: 
the series of 19 years which I used is not long enough. I 
regret that 1 have been unable to be ready in time with a 

Range of Meteorological and Magnetic Phenomena. 65 

longer series. But I fully expect that when this is quite 
computed these doubts will be cleared away. 

There is, however, another class of doubts which must 
perhaps be allowed as yet to stand over. It will be seen that 
between the maxima marked 15 and 16 I suggest for the 
upper curves two minima and an unimportant maximum 
between them, while lower down I have only been able to 
trace a single minimum. This may of course be consistent 
with the facts. Eventually for one phenomenon a certain 
maximum may become so slight that it can no longer be 
detected, at least not by our present methods ; and thus the 
two minima before and after it may merge into one, although 
other phenomena show two minima. 

Another fact of the same nature occurs perhaps between the 
July and August maxima, between the vertical lines 9 and 10. 
The dotted line between these two is much more crooked than 
the lines 9 and 10 seem to warrant. On the other hand, 
nearly all the lines show very slight indications that this 
minimum is indeed a double one separated by a very slight 
maximum. But this feature is so faint that I did not feel jus- 
tified for the present in drawing the vertical lines accordingly. 
Again the line 17 is a doubtful one. Of course this is 
partly only a technical difficulty. Where the curves are so 
near together as in this part of the diagram, the vertical lines 
must seem — not really be — -more crooked than they are in the 
summer part of the diagram, where they expand to such a 
length. Still I have a feeling that line 17 is also in reality 
composed of two maxima, of which one, in December, ought 
to have the number 17, and another, chiefly in January, the 
new number 18. 

I might of course multiply instances like these, but I feel 
that this is at present useless. The seventeen lines which I 
have numbered will, I think, be found in future to be a near 
approximation to truth for Western Europe so far as the most 
remarkable features go. But numberless other subsidiary 
vertical lines may eventually be filled in in course of time 
when more and better material shall be at hand than I could 
avail myself of as yet. 

But this, I think, is the result of this part of the work : 
There is one potent cause which for a large part rules all 
meteorological and magnetic phenomena, and influences them 
all in a similar way nearly simultaneously. 

1 do not of course pretend to teach a new doctrine in these 
words, but I think I have shown a new proof of it. And 
also how largely the simple method which 1 have used may in 
future contribute to the solution of meteorological problems. 
Phil. Mag. S. 5. Vol. 47. No. 284. Jan. 1899. F 

66 Lord Kelvin on the Age of the 

V. The Age of the Earth as an Abode fitted for Life. 
By the Eight Hon. Lord Kelvin, G.C.V.O. * 

§ 1. rilHE age of the earth as an abode fitted for life is cer- 
JL tainly a subject which largely interests mankind in 
general. For geology it is of vital and fundamental impor- 
tance — as important as the date of the battle of Hastings is 
for English history — yet it was very little thought of by 
geologists of thirty or forty years ago ; how little is illus- 
trated by a statement f, which I will now read, given originally 
from the presidential chair of the Geological Society by 
Professor Huxley in 1869, when for a second time, after a 
seven years' interval, he was president of the Society. 

" I do not suppose that at the present day any geologist would he found 
... to deny that the rapidity of the rotation of the earth may be 
diminishing, that the sun may be waxing dim, or that the earth itself may 
be cooling. Most of us, I suspect, are Gallios, ' who care for none of 
these things/ being of opinion that, true or fictitious, they have made no 
practical difference to the earth, during the period of which a record is 
preserved in stratified deposits." 

§ 2. I believe the explanation of how it was possible for 
Professor Huxley to say that he and other geologists did not 
care for things on which the age of life on the earth essen- 
tially depends, is because he did not know that there was 
valid foundation for any estimates worth considering as to 
absolute magnitudes. If science did not allow us to give any 
estimate whatever as to whether 10,000,000 or 10,000,000,000 
years is the age of this earth as an abode fitted for life, then 
1 think Professor Huxley would have been perfectly right in 
saying that geologists should not trouble themselves about 
it, and biologists should go on in their own way, not en- 
quiring into things utterly beyond the power of human 
understanding and scientific investigation. This would have 
left geology much in the same position as that in which 
English history would be if it were impossible to ascertain 
w T hether the battle of Hastings took place 800 years ago, or 
800 thousand years ago, or 800 million years ago. If it were 
absolutely impossible to find out which of these periods is 
more probable than the other, then I agree we might be 
Gallios as to the date of the Norman Conquest. But a 

* Communicated by the Author, being the 1897 Annual Address of the 
Victoria Institute with additions written at different times from June 
1897 to May 1898. 

t In the printed quotations the italics are mine in every case, not so 
the capitals in the quotation from Page's Text-book, 

Earth as an Abode fitted for Life. 67 

change took place just about the time to which I refer, and 
from then till now geologists have not considered the question 
of absolute dates in their science as outside the scope of their 

§ 3. I may be allowed to read a few extracts to indicate 
how geological thought was expressed in respect to this 
subject, in various largely used popular text-books, and in 
scientific writings which were new in 1868, or not so old as 
to be forgotten. I have several short extracts to read and 
I hope you will not find them tedious. 

The first is three lines from Darwin's " Origin of Species," 
1859 Edition, p. 287. 

" In all probability a far longer period than 300,000,000 years has 
elapsed since the latter part of tlie secondary period." 

Here is another still more important sentence, which I read 
to you from the same book : — 

" He who can read Sir Charles Lyell's grand work on the Principles 
of Geology, which the future historian will recognize as having produced 
a revolution in natural science, yet does not admit how incomprehensibly 
vast have been the past periods of time, may at once close this volume" 

1 shall next read a short statement from Page's ' Advanced 
Students' Text-Book of Geology,' published in 1859 : — 

" Again where the FORCE seems unequal to the result, the student 
should never lose sight of the element TIME : an element to ivhich we can 
set no bounds in the past, any more than we know of its limit in the 
future." _ 

" It will be seen from this hasty indication that there are two great 
schools of geological causation -the one ascribing every result to the 
ordinary operations of Nature, combined with the element of unlimited 
time, the other appealing to agents that operated during the earlier 
epochs of the world with greater intensity, and also for the most part 
over wider areas. The former belief is certainly more in accordance with 
the spirit of right philosophy, though it must be confessed that many 
problems in geology seem to find their solution only through the 
admission of the latter hypothesis." 

§ 4. I have several other statements which I think you may 
hear with some interest. Dr. Samuel Haughton, of Trinity 
College, Dublin, in his ' Manual of Geology/ published in 
1865, p. 82, says :• — 

"The infinite time of the geologists is in the past ; and most of their 
speculations regarding this subject seem to imply the absolute infinity of 
time, as if the human imagination was unable to grasp the period of time 
requisite for the formation of a few inches of sand or feet of mud, and its 
subsequent consolidation into rock." (This delicate satire is certainly 
not overstrained.) 

" Professor Thomson has made an attempt to calculate the length of 
time during which the sun can have gone on burning at the present rate, 
and has come to the following conclusion : — ' It seems, on the whole, most 


8 Lord Kelvin on the Age of the 

probable that the sun has not illuminated the earth for 100,000,000 years, 
and almost certain that he has not done so for 500,000,000 years. As for 
the future, we may say with equal certainty, that the inhabitants of the 
earth cannot continue to enjoy the light and heat essential to their life 
for many million years longer, unless new sources, now unknown to us, 
are prepared in the great storehouse of creation." 

I said that in the sixties and I repeat it now ; but with 
charming logic it is held to be inconsistent with a later state- 
ment that the sun has not been shining 60,000,000 years ; 
and that both that and this are stultified by a still closer 
estimate which says that probably the sun has not been 
shining for 30,000,000 years ! And so my efforts to find some 
limit or estimate for Geological Time have been referred to 
and put before the public, even in London daily and weekly 
papers, to show how exceedingly wild are the wanderings 
of physicists, and how mutually contradictory are their con- 
clusions, as to the length of time which has actually passed 
since the early geological epochs to the present date. 

Dr. Haughton further goes on — 

" This result (100 to 500 million years) of Professor Thomson's, although 
very liberal in the allowance of time, has offended geologists, because, having 
been accustomed to deal with time as an infinite quantity at their disptosal, 
they feel naturally embarrassment and alarm at any attempt of the science 
of Physics to place a limit upon their speculations. It is quite possible 
that even a hundred million of years may be greatly in excess of the 
actual time during which the sun's heat has remained constant." 

§ 5. Dr. Haughton admitted so much with a candid open 
mind; but he went on to express his own belief (in 1865) thus: — 

" Although I have spoken somewhat disrespectfully of the geological 
calculus in my lecture, yet I believe that the time during which organic 
life has existed on the earth is practically infinite, because it can be shown 
to be so great as to be inconceivable by beings of our limited intelligence." 

Where is inconceivableness in 10,000,000,000 ? There is 
nothing inconceivable in the number of persons in this room, 
or in London. We get up to millions quickly. Is there any- 
thing inconceivable in 30,000,000 as the population of 
England, or in 38,000,000 as the population of Great Britain 
and Ireland, or in 352,704,863 as the population of the 
British Empire ? Not at all. It is just as conceivable as half 
a million years or 500 millions. 

§ 6. The following statement is from Professor Jukes's 
' Students' Manual of Geology : ' — 

" The time required for such a slow process to effect such enormous 
results must of course be taken to be inconceivably great. The word 
' inconceivably ' is not here used in a vague but in a literal sense, to 
indicate that the lapse of time required for the denudation that has 
produced the present surfaces of some of the older rocks, is vast beyond 
any idea of time which the human mind is capable of conceiving." 

Earth as an Abode fitted for Life. 69 

u Mr. Darwin, in Ms admirably reasoned book on the origin of species, 
so full of information and suggestion on all geological subjects, estimates 
the time required for the denudation of the rocks of the Weald of Kent, 
or the erosion of spac3 between the ranges of chalk hills, known as the 
North and South Downs, at three hundred millions of years. The 
grounds for forming this estimate are of course of the vaguest de- 
scription. It may be possible, perhaps, that the estimate is a hundred 
times too great, and that the real time elapsed did not exceed three 
million years, but, on the other hand, it is just as likely that the time 
which actually elapsed since the first commencement of the erosion till it 
was nearly as complete as it now is, was really a hundred times greater 
than his estimate, or thirty thousand millions of years." 

§ 7. Thus Jukes allowed estimates of anything from 
3 millions to 30,000 millions as the time which actually 
passed during the denudation of the Weald. On the other 
hand Professor Phillips in his Rede lecture to the University 
of Cambridge (18(30), decidedly prefers one inch per annum 
to Darwin's one inch per century as the rate of erosion : and 
says that most observers would consider even the one inch 
per annum too small for all but the most invincible coasts ! 
He thus, on purely geological grounds, reduces Darwin's 
estimate of the time to less than one one-hundredth. And, 
reckoning the actual thicknesses of all the known geological 
strata of the earth, he finds 96 million years as a possible 
estimate for the antiquity of the base of the stratified rocks ; 
but he gives reasons for supposing that this may be an over- 
estimate, and he finds that from stratigraphical evidence 
alone, we may regard the antiquity of life on the earth as 
possibly between 38 millions and 96 millions of years. 
Quite lately a very careful estimate of the antiquity of 
strata containing remains of life on the earth has been 
given by Professor Sollas, of Oxford, calculated according 
to stratigraphical principles which had been pointed out by 
Mr. Alfred Wallace. Here it is * : — " So far as I can at 
present see, the lapse of time since the beginning of the 
Cambrian system is probably less than 17,000,000 years, 
even when computed on an assumption of uniformity, which 
to me seems contradicted by the most salient facts of geology. 
Whatever additional time the calculations made on physical 
data can afford us, may go to the account of pre-Cambrian 
deposits, of which at present we know too little to serve for 
an independent estimate." 

§ 8. In one of the evening Conversaziones of the British 
Association during its meeting at Dundee in 1867 I had a 
conversation on geological time with the late Sir Andrew 
Ramsay, almost every word of which remains stamped on 

* "The Age of the Earth," < Nature,' April 4th, 1895. 

70 Lord Kelvin on the Age of the 

my mind to this day. We had been hearing a brilliant and 
suggestive lecture by Professor (now Sir Archibald) Geikie 
on the geological history of the actions by which the existing 
scenery of Scotland was produced. I asked Ramsay how long 
a time he allowed for that history. He answered that he 
could suggest no limit to it. I said, " You don't suppose 
things have been going on always as they are now ? You 
don't suppose geological historvhas run through 1,000,000,000 
years?" "Certainly I do/' " 10,000,000,000 years V } ' ; Yes." 
" The sun is a finite body. You can tell how many tons it is. 
Do you think it has been shining on for a million million 
years ? " "I am as incapable of estimating and under- 
standing the reasons which you physicists have for limiting 
geological time as you are incapable of understanding the 
geological reasons for our unlimited estimates." I answered, 
" You can understand physicists' reasoning perfectly if you 
give your mind to it." I ventured also to say that physicists 
were not wholly incapable of appreciating geological diffi- 
culties ; and so the matter ended, and we had a friendly 
agreement to temporarily differ. 

§ 9. In fact, from about the beginning of the century till 
that time (1867), geologists had been nurtured in a philosophy 
originating with the Huttonian system: much of it substantially 
very good philosophy, but some of it essentially unsound and 
misleading : witness this, from Playfair, the eloquent and able 
expounder of Hutton : — 

"How often these vicissitudes of decay arid renovation have been 
repeated is not for us to determine ; they constitute a series of which as 
the author of this theory has remarked, we neither see the beginning nor 
the end ; a circumstance that accords well with what is known concerning 
other parts of the economy of the world. In the continuation of the 
different species of animals and vegetables that inhabit the earth, we 
discern neither a beginning nor an end : in the planetary motions where 
geometry has carried the eye so far both into the future 'and the past we 
discover no mark either of the commencement or the termination of the 
present order." 

§ 10. Led by Hutton and Playfair, Lyell taught the 
doctrine of eternity and uniformity in geologv : and to 
explain plutonic action and underground heat, invented a 
thermo-electric " perpetual " motion on which, in the vear 
1862, in my paper on the " Secular Cooling of the Earth"* 
published in the " Transactions of the Royal Societv of Edin- 
burgh/ I commented as follows : — 

* Reprinted in Thomson and Tait, ' Treatise on Natural Philosophy,' 
1st and 2nd Editions, Appendix D (g). 

Earth as an Abode fitted for Life. 71 

"To suppose, as Lyell, adopting the chemical hypothesis, has done*, 
that the substances, combining together, may be again separated electro- 
lytically by thermo-electric currents, due to the heat generated by their 
combination, and thus the chemical action and its heat continued in an 
endless cycle, violates the principles of natural philosophy in exactly the 
same manner, and to the same degree, as to believe that a clock con- 
structed with a self-winding movement may fulfil the expectations of its 
ingenious inventor by going for ever." 

It was only by sheer force of reason that geologists have 
been compelled to think otherwise, and to see that there was 
a definite beginning, and to look forward to a definite end, 
of this world as an abode fitted for life. 

§ 11. It is curious that English philosophers and writers 
should not have noticed how Newton treated the astro- 
nomical problem. Play fair, in what I have read to yon, speaks 
of the planetary^ system as being absolutely eternal, and 
unchangeable : having had no beginning and showing no 
signs of progress towards an end. He assumes also that the 
sun is to go on shining for ever, and that the earth is to go on 
revolving round it for ever. He quite overlooked Laplace's 
nebular theory ; and he overlooked Newton's counterblast 
to the planetary " perpetual motion.'" Newton, commenting 
on his own f First Law of Motion,' says, in his terse Latin, 
which I will endeavour to translate, a But the greater bodies 
of planets and comets moving in spaces less resisting, keep 
their motions longer," That is a strong counterblast against 
any idea of eternity in the planetary system. 

§ 12. I shall now, without further preface, explain, and I 
hope briefly, so as not to wear out your patience, some of the 
arguments that I brought forward between 1862 and 1869, 
to show strict limitations to the possible age of the earth as 
an abode fitted for life. 

Kant f pointed out in the middle of last century, what had 
not previously been discovered by mathematicians or physical 
astronomers, that the frictional resistance against tidal cur- 
rents on the earth's surface must cause a diminution of the 
earth's rotational speed. This really great discovery in 

* 'Principles of Geology,' chap. xxxi. ed. 1853. 

f In an essay first published in the Koenigsberg Nackrichten, 1754, 
Nos. 23, 24 ; having been written with reference to the offer of a prize by 
the Berlin Academy of Sciences in 1754. Here is the title-page, in full, 
as it appears in vol. vi. of Kant's Collected Works, Leipzig, 1839 : — 
Untersuchung der Frage : Ob die Erde in ihrer Umdrenung um die Achse, 
wodurch sie die Abwechselung des Tages und der Nacht hervorbringt, 
einige Veranderung seit den ersten Zeiten ihres Ursprunges erlitten habe, 
welches die Ursache davon sei, und woraus man sich ihrer versichern 
konne ? welche von der Koniglichen Akademie der Wissenschaften zu 
Berlin zum Preise aufgegeben worden, 1754. 

72 Lord Kelvin on the Age of the 

Natural Philosophy seems to have attracted very little 
attention, — indeed to have passed quite unnoticed, — among 
mathematicians, and astronomers, and naturalists, until about 
1840, when the doctrine of energy began to be taken to 
heart. In 1866, Delaunay suggested that tidal retardation 
of the earth's rotation was probably the cause of an out- 
standing acceleration of the moon's mean motion reckoned 
according to the earth's rotation as a timekeeper found by 
Adams in 1853 by correcting a calculation of Laplace which 
had seemed to prove the earth's rotational speed to be uni- 
form *. Adopting Delaunay's suggestion as true, Adams, in 
conjunction with Professor Tait and myself, estimated the 
diminution of the earth's rotational speed to be such that 
the earth as a timekeeper, in the course of a century, would 
get 22 seconds behind a thoroughly perfect watch or clock 
rated to agree with it at the beginning of the century. 
According to this rate of retardation the earth, 7,200 million 
years ago, would have been rotating twice as fast as now : 
and the centrifugal force in the equatorial regions would 
have been four times as great as its present amount, which 
is 2J9 of gravity. At present the radius of the equatorial 
sea-level exceeds the polar semi-diameter by 21-J- kilometres, 
which is, as nearly as the most careful calculations in the 
theory of the earth's figure can tell us, just what the excess 
of equatorial radius of the surface of the sea all round would 
be if the whole material of the earth were at present liquid 
and in equilibrium under the influence of gravity and centri- 
fugal force with the present rotational speed, and £ of what 
it would be if the rotational speed were twice as great. 
Hence, if the rotational speed had been twice as great as 
its present amount when consolidation from approximately 
the figure of fluid equilibrium took place, and if the solid 
earth, remaining absolutely rigid, had been gradually slowed 
down in the course of millions of years to its present speed 
of rotation, the water would have settled into two circular 
oceans round the two poles : and the equator, dry all round, 
would be 64*5 kilometres above the level of the polar sea 
bottoms. This is on the supposition of absolute rigidity of 
the earth after primitive consolidation. There would, in 
reality, have been some degree of yielding to the gravitational 
tendency to level the great gentle slope up from each pole to 
equator. But if the earth, at the time of primitive consolida- 

* ' Treatise on Natural Philosophy ' (Thomson and Tait), §830, ed. 1, 
1867, and later editions ; also ' Popular Lectures and Addresses/ vol. ii. 
(Kelvin), ' Geological Time,' being a reprint of an article communicated 
to the Glasgow Geological Society, February 27th, 1868. 

Earth as an Abode fitted for Life. 73 

tion, had bean rotating twice as fast as at present, or even 
20 per cent, faster than at present, traces of it's present figure 
must have been left in a great preponderance of land, and 
probably no sea at all, in the equatorial regions. Taking 
into account all uncertainties, whether in respect to Adams' 
estimate of the rate of frictional retardation of the earth's 
rotatory speed, or to the conditions as to the rigidity of the 
earth once consolidated, we may safely conclude that the 
earth was certainly not solid 5,000 million years ago, and 
was probably not solid 1,000 million years ago*. 

§ 13. A second argument for limitation of the earth's age, 
which was really my own first argument, is founded on the 
consideration of underground heat. To explain a first rough 
and ready estimate of it I shall read one short statement. 
It is from a very short paper that I communicated to the 
Eoyal Society of Edinburgh on the 18th December, 1865, 
entitled, " The Doctrine of Uniformity in Geology brief! y 

" The ' Doctrine of Uniformity ' in Geology, as held by many of the 
most eminent of British Geologists, assumes that the earth's surface and 
upper crust have been nearly as they are at present in temperature, and 
other physical qualities, during millions of millions of years. But the heat 
which we knoio, by observation, to be now conducted out of the earth yearly 
is so great, that if this action had been going on with any approach to 
uniformity for 20,000 million years, the amount of heat lost out of the 
earth would have been about as much as would heat, by 100° C, a 
quantity of ordinary surface rock of 100 times the earth's bulk. This 
would be more than enough to melt a mass of surface rock equal in bulk 
to the whole earth. No hypothesis as to chemical action, internal fluidity, 
effects of pressure at great depth, or possible character of substances in 
the interior of the earth, possessing the smallest vestige of probability, 
can justify the supposition that the earth's upper crust has remained 
nearly as it is, while from the whole, or from any part, of the earth, so 
great a quantity of heat has been lost." 

§ 14. The sixteen words which I have emphasized in read- 
ing this statement to you (italics in the reprint) indicate the 
matter-of-fact foundation for the conclusion asserted. This 
conclusion suffices to sweep away the whole system of geolo- 
gical and biological speculation demanding an " inconceiv- 
ably " great vista of past time, or even a few thousand million 
years, for the history of life on the earth, and approximate 
uniformity of plutonic action throughout that time ; which, 
as we have seen, was very generally prevalent thirty years 

* " The fact that the continents are arranged along meridians, rather 
than in an equatorial belt, affords some degree of proof that the consoli- 
dation of the earth took place at a time when the diurnal rotation differed 
but little from its present value. It is probable that the date of consoli- 
dation is considerably more recent than a thousand million years ago." — 
Thomson and Tait, 'Treatise on Natural Philosophy,' 2nd ed., 1883, § 830. 

74 Lord Kelvin on the Age of the 

ago among British Geologists and Biologists; and which, I 
must say, some of our chiefs of the present day have not yet 
abandoned. Witness the Presidents of the Geological and 
Zoological Sections of the British Association at its meetings 
of 1893 (Nottingham), and of 1896 (Liverpool). 

Mr. Teall : Presidential Address to the Geological Section, 1 893. 
" The good old British ship ' Uniformity,' built by Hutton and refitted 
by Lyell, has won so mauy glorious victories in the past, and appears still 
to be in such excellent righting trim, that I see no reason why she should 
haul down her colours either to ' Catastrophe' or ' Evolution.' Instead, 
therefore, of acceding to the request to ' hurry up ' we make a demand 
for more time." 

Professor Poulton : Presidential Address to the Zoological Section, 1896. 
" Our argument does not deal with the time required for the origin of 
life, or for the development of the lowest beings with which we are 
acquainted from the first formed beings, of which we know nothing. 
Both these processes may have required an immensity of time ; but as we 
know nothing whatever about them and have as yet no prospect of 
acquiring any information, we are compelled to confine ourselves to as 
much of the process of evolution as we can infer from the structure of 
living and fossil forms — that is, as regards animals, to the development 
of the simplest into the most complex Protozoa, the evolution of the 
Metazoa from the Protozoa, aud the branching of the former into its 
numerous Phyla, with all their Classes, Orders, Families, Genera, and 
Species. But we shall find that this is quite enough to necessitate a very 
large increase in the time estimated by the geologist." 

§ 15. In my own short paper from which I have read you 
a sentence, the rate at which heat is at the present time lost 
from the earth by conduction outwards through the upper 
crust, as proved by observations of underground temperature 
in different parts of the world, and by measurement of the 
thermal conductivity of surface rocks and strata, sufficed to 
utterly refute the Doctrine of Uniformity as taught by Hutton, 
Lyell, and their followers ; which was the sole object of that 

§ 16. In an earlier communication to the Royal Society 
of Edinburgh *, I had considered the cooling of the earth 
due to this loss of heat ; and by tracing backwards the 
process of cooling had formed a definite estimate of the 
greatest and least number of million years which can 
possibly have passed since the surface of the earth was 
everywhere red hot. I expressed my conclusion in the 
following statement f : — 

* " On the Secular Cooling of the Earth," Trans. Roy. Soc. Edinburgh, 
vol. xxiii. April 28th, 1862, reprinted in Thomson and Tait, vol. iii. 
pp. 468-485, and Math, and Phys. Papers, art. xciv. pp, 295-311. 

t " On the Secular Cooling of the Earth," Math, and Phys. Papers, 
vol. iii. § 11 of art. xciv. 

Earth as an Abode fitted for Life. 75 

" We are very ignorant as to the effects of high, temperatures in altering 
the conductivities and specific heats and melting temperatures of rocks, 
and as to their latent heat of fusion. We must, therefore, allow very- 
wide limits in such an estimate as I have attempted to make ; but I think 
we may with much probability sa} r that the consolidation cannot have 
taken place less than 20 million years ago, or we should now have more 
underground heat than we actually have ; nor more than 400 million years 
ago, or we should now have less underground heat than we actually have. 
That is to say, I conclude that Leibnitz's epoch of emergence of the 
consistentior status [the consolidation of the earth from red hot or white 
hot molten matter] was probably between those dates." 

§ 17. During the 35 years which have passed since I gave 
this wide-ranged estimate, experimental investigation has 
supplied much of the knowledge then wanting regarding the 
thermal properties of rocks to form a closer estimate of the 
time which has passed since the consolidation of the earth, 
and we have now good reason for judging that it was more 
than 20 and less than 40 million years ago ; and probably 
much nearer 20 than 40, 

§ 18. Twelve years ago, in a laboratory established by 
Mr. Clarence King in connexion with the United States 
Geological Survey, a very important series of experimental 
researches on the physical properties of rocks at high 
temperatures was commenced by Dr. Carl Barus, for the 
purpose of supplying trustworthy data for geological theory. 
Mr, Clarence King, in an article published in the 'American 
Journal of Science ; *, used data thus supplied, to estimate 
the age of the earth more definitely than was possible for me 
to do in 1862, with the very meagre information then available 
as to the specific heats, thermal conductivities, and tempera- 
tures of fusion, of rocks. I had taken 7000° F. (3781° C.) 
as a high estimate of the temperature of melting rock. Even 
then I might have taken something between 1000° C. and 
2000° C. as more probable, but I was most anxious not to 
under-estimate the age of the earth, and so I founded my 
primary calculation on the 7000° F. for the temperature of 
melting rock. We know now from the experiments of Carl 
Barus f that diabase, a typical basalt of very primitive 
character, melts between 1100° C. and 1170°, and is tho- 
roughly liquid at 1200°. The correction from 3871° C. to 
1200° or 1/3*22 of that value, for the temperature of solidifi- 
cation, would, with no other change of assumptions, reduce 
my estimate of 100 million to 1/(3*22) 2 of its amount, or a 
little less than 10 million years ; but the effect of pressure 
on the temperature of solidification must also be taken into 

* * On the Age of the Earth,' vol. xlv. January 1893. 
t Phil. Mag. 1893, first half-year, pp. 186, 187, 301-305. 

76 Lord Kelvin on the Age of the 

account, and Mr. Clarence King, after a careful scrutiny of 
all the data given him for this purpose by Dr. Barus, concludes 
that without further experimental data " we have no warrant 
for extending the earth's age beyond 24 millions of years." 

§ 19. By an elaborate piece of mathematical book-keeping 
1 have worked out the problem of the conduction of heat 
outwards from the earth, with specific heat increasing up to 
the melting-point as found by Riicker and Roberts-Austen 
and by Barus, but with the conductivity assumed constant ; 
and, by taking into account the augmentation of melting- 
temperature with pressure in a somewhat more complete 
manner than that adopted by Mr. Clarence King, I am not 
led to differ much from his estimate of 24 million years. 
But, until we know something more than we know at present 
as to the probable diminution of thermal conductivity with 
increasing temperature, which would shorten the time since 
consolidation, it would be quite inadvisable to publish any 
closer estimate. 

§ 20. All these reckonings of the history of underground 
heat, the details of which I am sure you do not wish me to 
put before you at present, are founded on the very sure 
assumption that the material of our present solid earth all 
round its surface was at one time a white hot liquid. The 
earth is at present losing heat from its surface all round from 
year to year and century to century. We may dismiss as 
utterly untenable any supposition such as that a few thousand 
or a few million years of the present regime in this respect 
was preceded by a few thousand or a few million years of 
heating from without. History, guided by science, is bound 
to find, if possible, an antecedent condition preceding every 
known state of affairs, whether of dead matter or of living 
creatures. Unless the earth was created solid and hot out 
of nothing, the regime of continued loss of heat must have 
been preceded by molten matter all round the surface. 

§ 21. 1 have given strong reasons * for believing that 
immediately before solidification at the surface, the interior 
was solid close up to the surface : except comparatively small 
portions of lava or melted rock among the solid masses of 
denser solid rock which had sunk through the liquid, and 
possibly a somewhat large space around the centre occupied 
by platinum, gold, silver, lead, copper, iron, and other dense 
metals, still remaining liquid under very high pressure. 

§ 22. I wish now to speak to you of depths below the 

* « On the Secular Cooling of the Earth," vol. iii. Math, and Phys. 
Papers, §§ 19-33. 

Earth as an Abode fitted for Life. 11 

great surface of liquid lava bounding the earth before 
consolidation ; and of mountain heights and ocean depths 
formed probably a few years after a first emergence of solid 
rock from the liquid surface (see § 24, below) , which must have 
been quickly followed by complete consolidation all round the 
globe. But I must first ask you to excuse my giving you all 
my depths, heights, and distances, in terms of the kilometre, 
being about six-tenths of that very inconvenient measure the 
English statute mile, which, with all the other monstrosities 
of our British metrical system, will, let us hope, not long- 
survive the legislation of our present Parliamentary session 
destined to honour the sixty years'' Jubilee of Queen Victoria's 
reign by legalising the French metrical system for the United 

§ 23. To prepare for considering consolidation at the surface 
let us go back to a time (probably not more than twenty 
years earlier as we shall presently see — § 24) when the solid 
nucleus was covered with liquid lava to a depth of several 
kilometres ; to fix our ideas let us say 40 kilometres (or 4 
million centimetres). At this depth in lava, if of specific 
gravity 2*5, the hydrostatic pressure is 10 tons weight (10 
million grammes) per square centimetre, or ten thousand 
atmospheres approximately. According to the laboratory 
experiments of Clarence King and Carl Barus * on Diabase, 
and the thermodynamic theory f of my brother, the late 
Professor James Thomson, the melting temperature of 
diabase is 1170° C. at ordinary atmospheric pressure, and 
would be 1420° under the pressure of ten thousand atmo- 
spheres, if the rise of temperature with pressure followed the 
law of simple proportion up to so high a pressure. 

§ 24. The temperature of our 40 kilometres deep lava 
ocean of melted diabase may therefore be taken as but 
little less than 1420° from surface to bottom. Its surface 
would radiate heat out into space at some such rate as two 
(gramme-water) thermal units Centigrade per square centi- 
metre per second J. Thus, in a year (31J million seconds) 

* Phil. Mag. 1893, first half-year, p. 306. 

t Trans. Roy. Soc. Edinburgh, Jan. 2, 1849 ; Cambridge and Dublin 
Mathematical Journal, Nov. 1850. Reprinted in Math, and Phys. Papers 
(Kelvin), vol. i. p. 156. 

% This is a very rough estimate which I have formed from consideration 
of J. T. Bottomley's accurate determinations in absolute measure of 
thermal radiation at temperatures up to 9^0° C. from platinum wire and 
from polished and blackened surfaces of various hinds in receivers of air- 
pumps exhausted down to one ten-millionth of the atmospheric pressure 
Phil, Trans. Roy. Soc, 1887 and 1893. 

78 Lord Kelvin on the Age of the 

63 million thermal units would be lost per square centimetre 
from the surface. This is, according to Carl Barus, very 
nearly equal to the latent heat of fusion abandoned by a 
million cubic centimetres of melted diabase in solidifying into 
the glassy condition (pitch-stone) which is assumed when 
the freezing takes place in the course of a few minutes. 
But, as found by Sir James Hall in his Edinburgh experi- 
ments *" of 100 years ago, when more than a few minutes is 
taken for the freezing, the solid formed is not a glass but a 
heterogeneous crystalline solid of rough fracture ; and if a 
few hours or days, or any longer time, is taken, the solid 
formed has the well known rough crystalline structure of 
basaltic rocks found in all parts of the world. Now Carl 
Barus finds that basaltic diabase is 14 per cent, denser than 
melted diabase, and 10 per cent, denser than the glass pro- 
duced by quick freezing of the liquid. He gives no data, 
nor do Riicker and Roberts-Austen, who have also experi- 
mented on the thermodynamic properties of melted basalt, 
give any data, as to the latent heat evolved in the consolida- 
tion of liquid lava into rock of basaltic quality. Guessing 
it as three times the latent heat of fusion of the diabase 
pitch- stone, I estimate a million cubic centimetres of liquid 
frozen per square centimetre per centimetre per three years. 
This would diminish the depth of the liquid at the rate of a 
million centimetres per three years, or 40 kilometres in twelve 

§ 25. Let us now consider in what manner this diminution 
of depth of the lava ocean must have proceeded, by the 
freezing of portions of it ; all having been at temperatures 
very little below the assumed 1420° melting temperature of 
the bottom, when the depth was 40 kilometreSo The loss of 
heat from the white-hot surface (temperatures from 1420° to 
perhaps 1380° in different parts) at our assumed rate of two 
(gramme-water Centigrade) thermal units per sq. cm. per 
sec. produces very rapid cooling of the liquid within a few 
centimetres of the surface (thermal capacity *36 per gramme, 
according to Barus) and in consequence great downward 
rushes of this cooled liquid, and upwards of hot liquid, 
spreading out horizontally in all directions when it reaches 
the surface. When the sinking liquid gets within perhaps 
20 or 10 or 5 kilometres of the bottom, its temperature f 

* Trans. Roy. Soc. Edinburgh. 

t Tlie temperature of the sinking liquid rock rises in virtue of the 
increasing pressure : but much less than does the freezing point of the 
liquid or of some of its ingredients. (See Kelvin, Math. andPhys. Papers, 
vol. iii. pp. 69, 70.) 

Earth as an Abode fitted for Life. 79 

becomes the freezing-point as raised by the increased 
pressure ; or, perhaps more correctly stated, a temperature 
at which some of its ingredients crystallize out of it. Hence, 
beginning a few kilometres above the bottom, we have a 
snow shower of solidified lava or of crystalline flakes, or 
prisms, or granules of felspar, mica, hornblende, quartz, and 
other ingredients : each little crystal gaining mass and falling 
somewhat faster than the descending liquid around it, till it 
reaches the bottom. This process goes on until, by the 
heaping of granules and crystals on the bottom, our lava 
ocean becomes silted up to the surface. 

Probable Origin of Granite. (§§ 26, 27.) 

§ 26. Upon the suppositions we have hitherto made we 
have, at the stage now reached, all round the earth at the 
same time a red hot or white hot surface of solid granules or 
crystals with interstices filled by the mother liquor still 
liquid, but ready to freeze with the slightest cooling. The 
thermal conductivity of this heterogeneous mass, even 
before the freezing of the liquid part, is probably nearly 
the same as that of ordinary solid granite or basalt at a 
red heat, which is almost certainly * somewhat less than the 
thermal conductivity of igneous rocks at ordinary tempera- 
tures. If you wish to see for yourselves how quickly it 
would cool when wholly solidified take a large macadamising 
stone, and heat it red hot in an ordinary coal fire. Take it 
out with a pair of tongs and leave it on the hearth, or on a 
stone slab at a distance from the fire, and you will see that 
in a minute or two, or perhaps in less than a minute, it cools 
to below red heat. 

§ 27. Half an hour f after solidification reached up to the 
surface in any part of the earth, the mother liquor among the 
granules must have frozen to a depth of several centimetres 
below the surface and must have cemented together the 
granules and crystals, and so formed a crust of primeval 
granite, comparatively cool at its upper surface, and red hot 
to white hot, but still all solid, a little distance down ; 
becoming thicker and thicker very rapidly at first ; and 
after a few weeks certainly cold enough at its outer surface 
to be touched by the hand. 

* Proc. K. S., May 30, 1895. 

t Witness the rapid cooling of lava running red hot or white hot from 
a volcano, and after a few days or weeks presenting a black hard crust 
strong enough and cool enough to be walked over with impunity. 

80 Lord Kelvin on the Age of the 

Probable Origin of Basaltic Bock * (§§ 28, 29.) 

§ 28. We have hitherto left, without much consideration, 
the mother liquor among the crystalline granules at all 
depths below the bottom of our shoaling lava ocean. It was 
probably this interstitial mother liquor that was destined 
to form the basaltic rock of future geological time. What- 
ever be the shapes and sizes of the solid granules when first 
falling to the bottom, they must have lain in loose heaps 
with a somewhat large proportion of space occupied by 
liquid among them. But, at considerable distances down in 
the heap, the weight of the superincumbent granules must 
tend to crush corners and edges into fine powder. If the 
snow shower had taken place in air we may feel pretty sure 
(even with the slight knowledge which we have of the hard- 
nesses of the crystals of felspar, mica and hornblende, and 
of the solid granules of quartz) that, at a depth of 10 kilo- 
metres, enough of matter from the corners and edges of the 
granules of different kinds, would have been crushed into 
powder of various degrees of fineness, to leave an exceed- 
ingly small proportionate volume of air in the interstices 
between the solid fragments. But in reality the effective 
weight of each solid particle, buoyed as it was by hydrostatic 
pressure of a liquid less dense than itself by not more than 
20 or 15 or 10 per cent., cannot have been more than from 
about one-fifth to one-tenth of its weight in air, and there- 
fore the same degree of crushing effect as would have been 
experienced at 10 kilometres with air in the interstices, must 
have been experienced only at depths of from 50 to 100 kilo- 
metres below the bottom of the lava ocean. 

§ 29. A result of this tremendous crushing together of the 
solid granules must have been to press out the liquid from 
among them, as water from a sponge, and cause it to pass 
upwards through the less and less closely packed heaps of 
solid particles, and out into the lava ocean above the heap. 
But, on account of the great resistance against the liquid 
permeating upwards 30 or 40 kilometres through interstices 
among the solid granules, this process must have gone on 
somewhat slowly ; and, during all the time of the shoaling of 
the larva ocean, there may have been a considerable proportion 
of the whole volume occupied by the mother liquor among 
the solid granules, down to even as low as 50 or 100 kilo- 
metres below the top of the heap, or bottom of the ocean, at 

* See Addendum at end of Lecture, 

Earth as an Abode fitted for Life. 81 

each instant. When consolidation reached the surface, the 
oozing upwards of the mother liquor must have been still 
going on to some degree. Thus, probably for a few years 
after the first consolidation at the surface, not probably for 
as long as one hundred years, the settlement of the solid 
structure by mere mechanical crushing of the corners and 
edges of solid granules, may have continued to cause the 
oozing upwards of mother liquor to the surface through 
cracks in the first formed granite crust and through fresh 
cracks in basaltic crust subsequently formed above it. 

Leibnitz's Consistentior Status. 
§ 30. When this oozing everywhere through fine cracks 
in the surface ceases, we have reached Leibnitz's consistentior 
status ; beginning with the surface cool and permanently 
solid and the temperature increasing to 1150° 0. at 25 or 50 
or 100 metres below the surface. 

Probable Origin of Continents and Ocean Depths of 
the Earth. (§§ 31-37.) 

§ 31. If the shoaling of the lava ocean up to the surface 
had taken place everywhere at the same time, the whole sur- 
face of the consistent solid would be the dead level of the 
liquid lava all round, just before its depth became zero. On 
this supposition there seems no possibility that our present- 
day continents could have risen to their present heights, 
and that the surface of the solid in its other parts could have 
sunk down to their present ocean depths, during the twenty 
or twenty-five million years which may have passed since the 
consistentior status began or during any time however long. 
Rejecting the extremely improbable hypothesis that the conti- 
nents were built up of meteoric matter tossed from without, 
upon the already solidified earth, we have no other possible 
alternative than that they are due to heterogeneousness in 
different parts of the liquid which constituted the earth before 
its solidification. The hydrostatic equilibrium of the rotating 
liquid involved only homogeneousness in respect to density 
over every level surface (that is to say, surface perpendicular 
to the resultant of gravity and centrifugal force) : it required 
no homogeneousness in respect to chemical composition. Con- 
sidering the almost certain truth that the earth was built up of 
meteorites falling together, we may follow in imagination the 
whole process of shrinking from gaseous nebula to liquid lava 
and metals, and solidification of liquid from central regions 
outwards, without finding any thorough mixing up of dif- 
ferent ingredients, coming together from different directions 

Phil. Mag. S. 5. Vol. 47, No. 284. Jan. 1899. G 

82 Lord Kelvin on the Age of the 

of space — any mixing up so thorough as to produce even 
approximately chemical homogeneousness throughout every 
layer of equal density. Thus we have no difficulty in under- 
standing how even the gaseous nebula, which at one time 
constituted the matter of our present earth, had in itself a 
heterogeneousness from which followed by dynamical neces- 
sity Europe, Asia, Africa, America, Australia, Greenland, and 
the Antarctic Continent, and the Pacific, Atlantic, Indian, 
and Arctic Ocean depths, as we know them at present. 

§ 32. We may reasonably believe that a very slight degree 
of chemical heterogeneousness could cause great differences 
in the heaviness of the snow shower of granules and crystals 
on different regions of the bottom of the lava ocean when 
still 50 or 100 kilometres deep. Thus we can quite see how 
it may have shoaled much more rapidly in some places than 
in others. It is also interesting to consider that the solid 
granules, falling on the bottom, may have been largely 
disturbed, blown as it were into ridges (like rippled sand in 
the bed of a flowing stream, or like dry sand blown into 
sand-hills by wind) by the eastward horizontal motion which 
liquid descending in the equatorial regions must acquire, 
relatively to the bottom, in virtue of the earth's rotation. It 
is indeed not improbable that this influence may have been 
largely effective in producing the general configuration of 
the great ridges of the Andes and Rocky Mountains and of 
the West Coasts of Europe and Africa. It seems, however, 
certain that the main determining cause of the continents and 
ocean-depths was chemical differences, perhaps very slight 
differences, of the material in different parts of the great lava 
ocean before consolidation. 

§ 33. To fix our ideas let us now suppose that over some 
great areas such as those which have since become Asia, 
Europe, Africa, Australia, and America, the lava ocean had 
silted up to its surface, while in other parts there still were 
depths ranging down to 40 kilometres at the deepest. In 
a very short time, say about twelve years according to our 
former estimate (§ 24) the whole lava ocean becomes silted 
up to its surface. 

§ 34. We have not time enough at present to think out 
all the complicated actions, hydrostatic and thermodynamic, 
which must accompany, and follow after, the cooling of the 
lava ocean surrounding our ideal primitive continent. Bv 
a hurried view, however, of the affair we see that in virtue 
of, let us say, 15 per cent, shrinkage by freezing, the level 
of the liquid must, at its greatest supposed depth, sink six 
kilometres relatively to the continents : and thus the liquid 

Earth as an Abode fitted for Life. 83 

must recede from them ; and their bounding coast-lines must 
become enlarged. And just as water runs out of a sandbank, 
drying when the sea recedes from it on a falling tide, so 
rivulets of the mother liquor must run out from the edges of 
the continents into the receding lava ocean. But, unlike 
sandbanks of incoherent sand permeated by water remaining 
liquid, our uncovered banks of white-hot solid crystals, with 
interstices full of the mother liquor, will, within a few hours 
of being uncovered, become crusted into hard rock by cooling 
at the surface, and freezing of the liquor, at a temperature 
somewhat lower than the melting temperatures of any of the 
crystals previously formed. The thickness of the wholly 
solidified crust grows at first with extreme rapidity, so that 
in the course of three or four days it may come to be as 
much as a metre. At the end of a year it may be as much 
as 10 metres ; with, a surface, almost, or quite, cool enough 
for some kinds of vegetation. In the course of the first few 
weeks the regime of conduction of heat outwards becomes 
such that the thickness of the wholly solid crust, as long as 
it remains undisturbed, increases as the square root of the 
time ; so that in 100 years it becomes 10 times, in 25 million 
years 5000 times, as thick as it was at the end of one year : 
thus, from one year to 25 million years after the time of 
surface freezing, the thickness of the wholly solid crust might 
grow from 10 metres to 50 kilometres. These definite num- 
bers are given merely as an illustration ; but it is probable 
they are not enormously far from the truth in respect to what 
has happened under some of the least disturbed parts of the 
earth's surface. 

§ 35. We have now reached the condition described above 
in § 30, with only this difference, that instead of the upper 
surface of the whole solidified crust being level we have 
in virtue of the assumptions of §§ 33, 34, inequalities of 
6 kilometres from highest to lowest levels, or as much more 
than 6 kilometres as we please to assume it. 

§ 36. There must still be a small, but important, proportion 
of mother liquor in the interstices between the closely packed 
uncooled crystals below the wholly solidified crust. This 
liquor, differing in chemical constitution from the crystals, has 
its freezing-point somewhat lower, perhaps very largely lower, 
than the lowest of their melting-points. But, when we con- 
sider the mode of formation (§ 25) of the crystals from the 
mother liquor, we must regard it as still always a solvent 
ready to dissolve, and to redeposit, portions of the crystalline 
matter, when slight variations of temperature or pressure 
tend to cause such actions. Now as the specific gravity of 


84 Lord Kelvin on the Age of the 

the liquor is less, by something like 15 per cent., than the 
specific gravity of the solid crystals, it must tend to find its 
way upwards, and will actually do so, however slowly, until 
stopped by the already solidified impermeable crust, or until 
itself becomes solid on account of loss of heat by conduction 
outwards. If the upper crust were everywhere continuous 
and perfectly rigid the mother liquor must, inevitably, if 
sufficient time be given, find its way to the highest places of 
the lower boundary of the crust, and there form gigantic 
pockets of liquid lava tending to break the crust above it 
and burst up through it. 

§ 37. But in reality the upper crust cannot have been 
infinitely strong ; and, judging alone from what we know of 
properties of matter, we should expect gigantic cracks to 
occur from time to time in the upper crust tending to shrink 
as it cools and prevented from lateral shrinkage by the non- 
shrinking uncooled solid below it. When any such crack 
extends downwards as far as a pocket of mother liquor 
underlying the wholly solidified crust, we should have an 
outburst of trap rock or of volcanic lava just such as have 
been discovered by geologists in great abundance in many 
parts of the world. We might even have comparatively 
small portions of high plateaus of the primitive solid earth 
raised still higher by outbursts of the mother liquor squeezed 
out from below them in virtue of the pressure of large sur- 
rounding portions of the superincumbent crust. In any such 
action, due to purely gravitational energy, the centre of 
gravity of all the material concerned must sink, although 
portions of the matter may be raised to greater heights ; but 
we must leave these large questions of geological dynamics, 
having been only brought to think of them at all just now by 
our consideration of the earth, antecedent to life upon it. 

§ 38. The temperature to which the earth's surface cooled 
within a few years after the solidification reached it, must 
have been, as it is now, such that the temperature at which 
heat radiated into space during the night exceeds that re- 
ceived from the sun during the day, by the small difference 
due to heat conducted outwards from within *, One year 

* Suppose, for example, the cooling and thickening of the upper crust 
has proceeded so far, that at the surface and therefore approximately for 
a few decimetres below the surface, the rate of augmentation of tem- 
perature downwards is one degree per centimetre. Taking as a rough 
average -005 c.G.s. as the thermal conductivity of the surface rock, we 
should have for the heat conducted outwards •005 of a gramme water 
thermal unit centigrade per sq. cm. per sec. (Kelvin, Math, and Phys. 
Papers, vol. iii. p. 226). Hence if (ibid. p. 223) we take -g-^ as the 

Earth as an Abode fitted for Life. 85 

after the freezing of the granitic interstitial mother liquor at 
the earth's surface in any locality, the average temperature 
at the surface might be warmer, by 60° or 80° Cent., than if 
the whole interior had the same average temperature as the 
surface. To fix our ideas, let us suppose, at the end of one 
year, the surface to be 80° warmer than it would be with no 
underground heat : then at the end of 100 years it would be 
8° warmer, and at the end of 10,000 years it would be '8 of 
a degree warmer, and at the end of 25 million years it would 
be *016 of a degree warmer, than if there were no under- 
ground heat. 

§ 39. When the surface of the earth was still white-hot 
liquid all round, at a temperature fallen to about 1200° Cent., 
there must have been hot gases and vapour of water above 
it in all parts, and possibly vapours of some of the more 
volatile of the present known terrestrial solids and liquids, 
such as zinc, mercury, sulphur, phosphorus. The very rapid 
cooling which followed instantly on the solidification at the 
surface must have caused a rapid downpour of all the vapours 
other than water, if any there were ; and a little later, rain 
of water out of the air, as the temperature of the surface 
cooled from red heat to such moderate temperatures as 40° 
and 20° and 10° Cent., above the average due to sun heat 
and radiation into the aether around the earth. What that 
primitive atmosphere was, and how much rain of water fell 
on the earth in the course of the first century after consoli- 
dation, we cannot tell for certain ; but Natural History and 
Natural Philosophy give us some foundation for endeavours 
to discover much towards answering the great questions,— 
Whence came our present atmosphere of nitrogen, oxygen, 
and carbonic acid ? Whence came our present oceans and 
lakes of salt and fresh water ? How near an approximation 

radiational eniissivity of rock and atmosphere of gases and watery vapour 
above it radiating- heat into the surrounding vacuous space -(aether), we 
find 8000 X -005, or 40 degrees Cent, as the excess of the mean surface 
temperature above what it would be if no heat were conducted from 
within outwards. The present augmentation of temperature downwards 
may be taken as 1 degree Cent, per 27 metres as a rough average derived 
from observations in all parts of the earth where underground temperature 
has been observed. (See British Association Reports from 1868 to 1895. 
The very valuable work of this Committee has been carried on for these 
twenty-seven years with great skill, perseverance, and success, by 
Professor Everett, and he promises a continuation of his reports from 
time to time.) This with the same data for conductivity and radiational 
emissivity as in the preceding calculation makes 40°/2700 or 00118 u Cent. 
per centimetre as the amount by which the average temperature of the 
earth's surface is at present kept up by underground heat. 

86 Lord Kelvin on the Age of the 

to present conditions was realized in the first hundred cen- 
turies after consolidation of the surface ? 

§ 40. We may consider it as quite certain that nitrogen 
gas, carbonic acid gas, and steam, escaped abundantly in 
bubbles from the mother liquor of granite, before the primi- 
tive consolidation of the surface, and from the mother liquor 
squeezed up from below in subsequent eruptions of basaltic 
rock ; because all, or nearly all, specimens of granite arid 
basaltic rock, which have been tested by chemists in respect 
to this question *, have been found to contain, condensed in 
minute cavities within them, large quantities of nitrogen, 
carbonic acid, and water. It seems that in no specimen of 
granite or basalt tested has chemically free oxygen been dis- 
covered, while in many, chemically free hydrogen has been 
found ; and either native iron or magnetic oxide of iron in 
those which do not contain hydrogen. From this it might 
seem probable that there was no free oxygen in the primitive 
atmosphere, and that if there was free hydrogen, it was due 
to the decomposition of steam by iron or magnetic oxide of 
iron. Going back to still earlier conditions we might judge 
that, probably, among the dissolved gases of the hot nebula 
which became the earth, the oxygen all fell into combination 
with hydrogen and other metallic vapours in the cooling of 
the nebula, and that although it is known to be the most 
abundant material of all the chemical elements constituting 
the earth, none of it was left out of combination with other 
elements to give free oxygen in our primitive atmosphere. 

§ 41. It is, however, possible, although it might seem not 
probable, that there was free oxygen in the primitive atmo- 
sphere. With or without free oxygen, however, but with 
sunlight, we may regard the earth as fitted for vegetable life 
as now known in some species, wherever water moistened the 
newly solidified rocky crust cooled down below the tempera- 
ture of 80° or 70° of our present Centigrade thermometric 
scale, a year or two after solidification of the primitive lava 
had come up to the surface. The thick tough velvety coating 
of living vegetable matter, covering the rocky slopes under 
hot water flowing direct out of the earth at Banff (Canada) |, 
lives without help from any ingredients of the atmosphere 
above it, and takes from the water and from carbonic acid or 
carbonates, dissolved in it, the hydrogen and carbon needed 
for its own growth by the dynamical power of sunlight ; thus 

* See, for example, Tilden, Proc. R. S. February 4th, 1897. "On the 
Gases enclosed in Crystalline Rocks and Minerals." 

f Rocky Mountains Park of Canada, on the Canadian Pacific Railway. 

Earth as an Abode fitted for Life. 87 

leaving free oxygen in the water to pass ultimately into the 
air. Similar vegetation is found abundantly on the terraces 
of the Mammoth hot springs and on the beds of the hot water 
streams flowing from the Geysers in the Yellowstone National 
Park of the United States. This vegetation, consisting of 
confervas, all grows under flowing water at various tempera- 
tures, some said to be as high as 74° Cent. We cannot doubt 
but that some such confervas, if sown or planted in a rivulet 
or pool of warm water in the early years of the first century 
of the solid earth's history, and if favoured with sunlight, 
would have lived, and grown, and multiplied, and would have 
made a beginning of oxygen in the air, if there had been 
none of it before their contributions. Before the end of the 
century, if sun-heat, and sunlight, and rainfall, were suitable, 
the whole earth not under water must have been fitted for 
all kinds of land plants which do not require much or any 
oxygen in the air, and which can find, or make, place and 
soil for their roots on the rocks on which they grow ; and 
the lakes or oceans formed by that time must have been 
quite fitted for the life of many or ail of the species of water 
plants living on the earth at the present time. The moderate 
warming, both of land and water, by underground heat, 
towards the end of the century, would probably be favourable 
rather than adverse to vegetation, and there can be no doubt 
but that if abundance of seeds of all species of the present 
day had been scattered over the earth at that time, an im- 
portant proportion of them would have lived and multiplied 
by natural selection of the places where they could best 

§ 42. But if there was no free oxygen in the primitive 
atmosphere or primitive water, several thousands, possibly 
hundreds of thousands, of years must pass before oxygen 
enough for supporting animal life, as we now know it, was 
produced. Even if the average activity of vegetable growth 
on land and in water over the whole earth was, in those early 
times, as great in respect to evolution of oxygen as that of 
a Hessian forest, as estimated by Liebig* 50 years ago, or 
of a cultivated English hayfield of the present day, a very 
improbable supposition, and if there were no decay (erema- 
causis, or gradual recombination with oxygen) of the plants 
or of portions such as leaves falling from plants, the rate of 
evolution of oxygen, reckoned as three times the weight of 
the wood or the dry hay produced, would be only about 

* Liebig', ' Chemistry in its application to Agriculture and Physio- 
logy,' English, 2nd ed., edited by Playfair, 1842. 

88 Lord Kelvin on the Age of the 

6 tons per English acre per annum or 1^ tons per square 
metre per thousand years. At this rate it would take only 
1533 years, and therefore in reality a much longer time 
would almost certainly be required, to produce the 2*3 tons 
of oxygen which we have at present resting on every square 
metre of the earth's surface, land and sea*. But probably 
quite a moderate number of hundred thousand years may 
have sufficed. It is interesting at all events to remark that, 
at any time, the total amount of combustible material on the 
earth, in the form of living plants or their remains left dead, 
must have been just so much that to burn it all would take 
either the whole oxygen of the atmosphere, or the excess of 
oxygen in the atmosphere at the time, above that, if any, 
which there was in the beginning. This we can safely say, 
because we almost certainly neglect nothing considerable in 
comparison with what we assert when we say that the free 
oxygen of the earth's atmosphere is augmented only by 
vegetation liberating it from carbonic acid and water, in 
virtue of the power of sunlight, and is diminished only by 
virtual burning f of the vegetable matter thus produced. 
But it seems improbable that the average of the whole 
earth — dry land and sea-bottom — contains at present coal, 
or wood, or oil, or fuel of any kind originating in vegeta- 
tation, to so great an amount as "767 of a ton per square 
metre of surface ; which is the amount at the rate of one ton 
of fuel to three tons of oxygen, that would be required to 
produce the 2'3 tons of oxygen per square metre of surface, 
which our present atmosphere contains. Hence it seems 
probable that the earth's primitive atmosphere must have 
contained free oxygen. 

§ 43. Whatever may have been the true history of our 
atmosphere it seems certain that if sunlight was ready, the 
earth was ready, both for vegetable and animal life, if not 
within a century, at all events within a few hundred cen- 
turies after the rocky consolidation of its surface. But was 
the sun ready ? The well founded dynamical theory of the 
sun's heat carefully worked out and discussed by Helmholtz, 

* In our present atmosphere, in average conditions of barometer and 
thermometer we have, resting on each square metre of the earth's surface, 
ten tons total weight, of which 7-7 is nitrogen and 2*3 is oxygen. 

f This " virtual burning " includes ereuiacausis of decay of vegetable 
matter, if there is any eremacausis of decay without the intervention of 
microbes or other animals. It also includes the combination of a portion 
of the food with inhaled oxygen in the regular animal economy of pro- 
vision for heat and power. 

Earth as an Abode fitted for Life. 89 

Newcomb, and myself * says NO if the consolidation of the 
earth took place as long ago as 50 million years ; the solid 
earth mast in that case have waited 20 or 50 million years 
for the sun to be anything nearly as warm as he is at present. 
If the consolidation of the earth was finished 20 or 25 million 
years ago, the sun was probably ready, — though probably 
not then quite so warm as at present, yet warm enough 
to support some kind of vegetable and animal life on the 

§ 44. My task has been rigorously confined to what, 
humanly speaking, we may call the fortuitous concourse of 
atoms, in the preparation of the earth as an abode fitted for 
life, except in so far as 1 have referred to vegetation, as 
possibly having been concerned in the preparation of an 
atmosphere suitable for animal life as we now have it. 
Mathematics and dynamics fail us when Ave contemplate 
the earth, fitted for life but lifeless, and try to imagine the 
commencement of life upon it. This certainly did not take 
place by any action of chemistry, or electricity, or crystalline 
grouping of molecules under the influence of force, or by any 
possible kind of fortuitous concourse of atoms. We must 
pause, face to face with the mystery and miracle of the 
creation of living creatures. 

Addendum. — May 1898. 

Since this lecture was delivered I have received from 
Professor Roberts-Austen the following results of experiments 
on the melting-points of rocks which he has kindly made at 
my request : — 

Melting-point. Error. 

Felspar. . . 1520° C. ±30° 

Hornblende . about 1400° 

Mica . . . 1440° ±30° 

Quartz . . . 1775° +15° 

Basalt . . . about 880° 

These results are in conformity with what I have said in 
§§ 26-28 on the probable origin of granite and basalt, as 
they show that basalt melts at a much lower temperature 
than felspar, hornblende, mica, or quartz, the crystalline in- 
gredients of granite. In the electrolytic process for pro- 
ducing aluminium, now practised by the British Aluminium 

* See ' Popular Lectures and Addresses,' vol. i. pp. 376-429, par- 
ticularly page 397. 

90 Mr. D. L. Chapman on the 

Company at their Foyers works, alumina, of which the 
melting-point is certainly above 1700° C. or 1800° C, is 
dissolved in a bath of melted cryolite at a temperature of 
about 800° C. So we may imagine melted basalt to be a 
solvent for felspar, hornblende, mica, and quartz at tempera- 
tures much below their own separate melting-points ; and we 
can understand how the basaltic rocks of the earth may have 
resulted from the solidification of the mother liquor from 
which the crystalline ingredients of granite have been 

VI. On the Hate of Explosion in Gases. 
By D. L. Chapman, B.A. (Oxon.)*. 

THE object of the investigation of which an account is 
given in this paper is the discovery of formulae to 
express the maximum rates of explosion in gases and the 
maximum pressure in the explosive wave. 

The data which 1 propose to use are taken almost entirely 
from the Bakerian Lecture of 1893, on "The Rates of 
Explosion in Gases," by Prof. Dixon. The maximum 
velocities of explosion given below are in all cases those 
measured by Prof. Dixon or under his direction. Experi- 
mental conclusions only will be quoted ; for a complete 
account of the experiments themselves, the reader is referred 
to the above-mentioned paper, and to several papers which 
were subsequently published in the ' Journal of the Man- 
chester Literary and Philosophical Society ' and in the 
; Journal of the Chemical Society. 5 

Ignoring for the present all minor details connected with 
particular cases, which may be more conveniently discussed 
at a later stage, it is sufficient for our purpose to state at the 
outset that it has been established that the maximum velocity 
of explosion, in a mixture of definite composition and at fixed 
temperature and pressure, has a definite value, independent 
of the diameter of the tube when that diameter exceeds a 
certain limit. The relations existing between temperature 
and pressure and the velocity of explosion are such that an 
increase of temperature causes a fall in the velocity, whereas 
an increase of pressure has the reverse effect up to a certain 
limit, beyond which the velocity remains constant. 

For the suggestion that an explosion is in its character 

essentially similar to a sound-wave, we are also indebted to 

Prof. Dixon ; and there is little doubt that all subsequent 

advance must be made with, this suggestion as the leading 

* Communicated by Prof. Dixon, F.R.S. 

Rate of Explosion in Gases. 91 

idea *. Although Prof. Dixon's sound-wave formula has 
yielded such excellent results, he has pointed out the necessity 
of further a priori work in the subject. 

The Rate of Explosion for an Infinite Plane Wave. 

In the following attempt to establish a formula for the 
velocity of explosion, I have made certain assumptions which 
have not as yet received sufficient experimental confirmation ; 
hut they are, I think, justified by the results. For instance, 
it is assumed that, once the maximum velocity is reached, the 
front of the explosion wave is of such a character that we may 
suppose steady motion. This, as Prof. Schuster has pointed 
out in a note to the Bakerian Lecture, is not an impossibility 
when chemical change is taking place, since the implied 
relation between pressure and density is possible under such 
circumstances. This point, however, requires further investi- 
gation. The wave is assumed to be an infinite plane wave. 
This assumption is justified by the fact that the diameter of 
the tube is without influence on the found velocity. I propose 
to limit the term u explosive wave " to the space within which 
chemical change is taking place. This space is bounded by 
two infinite planes. On either side of the wave are the 
exploded and unexploded gases, which are assumed to have 
uniform densities and velocities. The statement that the 
exploded gas possesses uniform density and velocity for some 
distance behind the wave requires further justification, which 
can only be imperfectly given after a discussion of the general 

How the true explosive wave is actually generated in 
practice is a question without the scope of the present investi- 
gation. In order to avoid the discussion of this point, I 
shall substitute for it a physical conception, which, although 
unrealizable in practice, will render aid in illustrating the 
views here advanced. 

Let us suppose that the gas is enclosed in an infinite 
cylinder ABCD, provided with a piston E, and that the 
explosive wave XYZS has just started. The initial velocity 
of this wave will be small ; the initial pressure along the 
plane XS will also be small compared with that ultimately 
attained. As the wave proceeds in the direction AB, the 
piston E is supposed to follow it in such a manner that 

* In the earlier researches Berthelot's theory was accepted as a working 
hypothesis. It was only after the difficulties attending- the measurement 
of the rates of explosion in mixtures containing inert gases had been over- 
come that the inadequacy of Berthelot's theory became evident and the 
superiority of the sound-wave theory could be demonstrated. 

92 Mr. D. L. Chapman on the 

the pressure at EF is always kept equal to the pressure 
at XS. During this process the velocity of the wave will 
gradually increase, until ultimately its velocity will be 
uniform, its type constant, and the exploded gas within 
the area EXSF homogeneous. It is this ultimate steady 

Fig. 1. 

state alone which I propose to consider. During the process 
just described the velocity will of course constantly increase 
until it attains a maximum. After the velocity has become 
uniform, and the wave permanent in type, it is obvious that 
another permanent state may be reached in the following- 
way : — Suppose a piston is introduced immediately behind 
the permanent wave, and that this piston is made to move 
forward more rapidly than the previous one, the pressure and 
density behind the wave will thus be increased, and after a 
certain period of time another steady state will be reached. 
All this is equivalent to the statement that the permanent 
velocity of explosion is a function of the density of the 
exploded gas. 

I shall now proceed to prove the latter statement. 

Since the discussion is limited to the wave of permanent 
type, we may write down the condition of steady motion, 



where V and u are the velocities of the unexploded and 
exploded gas respectively, referred to coordinates moving 
with a velocity — V, and v and v are the volumes of a gram- 
equivalent of the unexploded and exploded gas. 

Take as an example cyanogen and oxygen, the explosion 
of which is represented by the equation 

C 2 N 2 + 2 2CO + N 2 . 

22-4 litres + 22'4 litres = 44*8 litres + 22*4 litres. 
52 grms. + 32 grins. = 56 grms. + 28 grms. 

Here v = 44'8 litres, and v is the volume of carbon monoxide 

Hate of Explosion in Gases. 93 

and nitrogen obtained from this immediately after the 

fi = gram-equivalent (in this case, 84 grms.). 

From (1) and the equations of motion, we obtain 

aV 2 
i ? -i>o= f rr(«o-v) (2)* 

v o 

This formula of Riemann assumes a relation to subsist 
between Y, p, and v at all points of the wave ; and from it 
the work performed by the wave during explosion may be 
calculated. • - 

Work performed by the gas 
fiY 2 


( v -v y+p {v-v ) 

For the purpose of testing this result, it maybe shown that 
the external work performed by the piston (fig. 1) is equal to 
the work performed on the gas together with the gain of 
kinetic energy. 

The work performed on the gas 

= fp - v o) 2 +Po( v o - «0 • 
The gain of kinetic energy 

_ (V-«)V 

it V 2 Yv 

The external work performed by the piston 
= p(v -v) 
_ fj,Y 2 

m {vo-v) 2 +p (v -v). 

.*. External work performed by the piston 

= gain of kinetic energy -f work performed on the gas. 

Assume that in the explosion n molecules become m mole- 
cules. For example, in the explosion of equal volumes of 

* Rayleigh's ' Sound/ vol. ii, j Schuster's note in the Bakerian Lecture 
on Explosions, 

94 Mr. D. L. Chapman on the 

cyanogen and oxygen, n is 2 and m is 3 : 

C 2 N 2 + 2 = 2CO + N 2 . 

(2 molecules) (3 molecules) 

I shall now calculate the energy lost when a gas is allowed 
to burn and the products of combustion are collected at the 
normal temperature and pressure. 

Assume that one of the gases is enclosed in the cylinder A 
and the other in the cylinder B (fig. 2). These gases are forced 

out, burned at C, cooled at D, and collected in the cylinder E. 
The gain of energy is the work performed by the pistons 
a and b ; and the loss of energy is the heat evolved at D, 
together with the work performed on the piston e. The total 
energy lost is the difference of these. The volume of gas in 
A and B is v ; therefore the work performed by the pistons 

a and b is p v Q . The volume of burnt gas is — - ; and 

therefore the work performed on the piston e is ■ ^° ° . 

The heat evolved at D is the heat of combustion at constant 
pressure ; call it h. Let the total energy lost =H. 


H = A+/w>o(~l) 

During an explosion the whole of this energy is retained 
by the gas, and in addition to this it gains an amount of 
energy equal to the work performed on the gas. 

The energy of the exploded gas is therefore given by the 

+ energy of exploded gas at N.T.P. 

n~Sf VYl 

= '1 + 15-2(0— VqY-PqV+PqVo- + energy at N,T.P. 

Rate of Explosion in Gases. 95 

If t is the normal temperature, and t the temperature of 
the gas after explosion, 

mO v (t— 1 ) + energy at N.T. P. = energy of exploded gas ; 

LLX 771 

/. mC v (t-to) = h + ^tv-Voy—poV+poVo-; 

.-- iS ^ n - +t , (8) 

But p = mRf, 

Also, from equation (2), we get 

pv = !— T (v — v)v+p v ; 
v o 

•' • 7T "! 7 > + ^-« (*> " V o) 2 - W + PoW^ i + mR«o 

= --2( v o-v)v+p v. . (4) 

This establishes a relation between V and v. The velocity 
of a permanent explosion is therefore a function of the density 
of the exploded gas. 

When an explosion starts its character and velocity are 
continually changing until it becomes a wave permanent in 
type and of uniform velocity. I think it is reasonable to 
assume that this wave — i. e. the wave of which the velocity 
has been measured by Prof. Dixon — is that steady wave 
which possesses minimum velocity ; for, once it has become a 
permanent wave with uniform velocity, no reason can be 
discovered for its changing to another permanent wave 
having a greater uniform velocity and a greater maximum 

This particular velocity may be discovered by eliminating v 
from the equations 

and dV 


It may be well to point out that under these circumstances 
the entropy of the exploded gas is a maximum. This may be 
easily shown thus : — 

96 Mr. J), L, Chapman on the 

The condition for maximum entropy is 

= dcj> = mC v f+£dv; 


mC v dt = —pdV) 
or dt _ p 

do mC v ' 

By differentiating (3), 
but from (a) and Riemann's equation 

therefore the condition of minimum velocity is equivalent to 
the condition of maximum entropy*. 

The following method of arriving at the approximation 

v= r ^ v was suggested by Prof. Schuster, who has 

shown that the method by which I arrived at the same result 
is inconclusive. 

Equation (4) arranged differently runs 

TJ^jv-VoY gV* _., r , , b 

where H does not contain v. 
Or putting R = C-C W , 

v )v=p v I 1 + q-J — H, 

^v^-t?o) rc P -c a/ . -i c P 

H— p v 


~ ^c! 

The complete expression ~r- =0 leads to a quadratic ex- 
pression for v. Hence there are two minima or maxima. 

* In any adiabatic change the entropy cannot decrease, and therefore 
it tends to become a maximum. 

Rate of Explosion in Gases. 97 


v=v , Y 2 =oo, 

Hence one minimum lies between these values*. 

If H is large compared to p v pp ? V will be a minimum or 

maximum when 

/ \ rOp— C„ Cp + 0„ "1 

is a maximum or minimum. 

Writing this F: F will be zero for 

G p — G v 

v = oTTc^ 

For v = + go it will be negative ; hence between the above 
values of v there will be a maximum of F or a minimum of V. 

2C v ^ = (C p -G v )v -(G p +G)v-(v-v )(C p + C v ) 

=2G P v -2(G p + G v )v. 

d 2 ¥ 
And yt * s a ^ wa y s negative ; hence F must be a maximum 


— u+o (5) 

By eliminating v from (4) and (5) we obtain the value 
of V 2 . This elimination leads to the result 

= -j-Qi [{{m-n)G p +mG v }G p t + (C p + C )/i], 

since p Q VQ=nRt . 

It is assumed throughout that the exploded gas behind the 
wave remains at constant temperature and pressure, and has 

* The other value of v obtained from the quadratic equation -j~ =0 is 

much larger than v , and gives to V a very small value. It has therefore 
no connexion with the wave we are considering. 

Phil. Mag. 8. 5. Vol. 47. No 284. Jan. 1899. H 

98 Mr. D. L. Chapman on the 

a uniform velocity. Therefore during the explosion momentum 
is generated by the moving piston. In an actual explosion in 
a tube not provided with a piston the whole mass of gas cannot 
move forward with this uniform velocity, for there would then 
be a vacuum at the end of the tube where the explosion started, 
and the pressure at that end would be zero, making it im- 
possible to account for the generation of momentum. There 
is, however, no need to assume that the whole exploded gas 
acquires a uniform velocity. In fact the velocity of the wave 
would be the same if it were followed by a layer of exploded 
gas of uniform density and velocity, and would be un affected 
by any subsequent disturbance which must take place behind 
the explosive wave. 

It is therefore necessary to prove that behind the explosive 
wave there is a layer of homogeneous gas. This evidently 
must he if any disturbance behind the wave can only move 
forward with a velocity less than that of the wave itself. 

The forward velocity of any disturbance in the exploded 
gas will be given by the sum of the velocity of the gas and 
the velocity of sound in the gas. 

The velocity of the gas 


v Vc„+cJ 


The velocity of sound 

_ /m&t Up 

In the complete expression for V 2 the first term may be 
here neglected. Also in the complete expression for t 
(equation (3)) the last three terms are small. We may 
therefore write 


^Y 2 

h +2^(v-Vo) 

Employing these values, the velocity of the gas becomes 

V— fere) 

Hate of Explosion in Gases. 99 

and the velocity of sound 

./ijg7 p "." 

.*. the velocity or sound + velocity of the gas 
_ /2RA C/+C„ 2 

"V ,, ■g*(c p +v v ) 

The velocity of explosion 


2R/i (,(V+0.) ! 
0/(0, + 0.) 

The latter is evidently greater than the former. Therefore 
the layer of uniform gas behind the wave will gradually 
become greater as the explosion proceeds. 

Calculation of the Rates of Explosion. 

In attempting to calculate the rates of explosion from the 
formula there is some doubt as to what value should be adopted 
for the specific heat at constant volume. This constant, has 
only been directly found at comparatively low temperatures. 
MM. Berthelot, Le Chatelier, and Mallard have made attempts 
to find the specific heats of the elementary gases and of carbon 
monoxide at high temperatures by measuring the pressure of 
explosion. Berthelot arrives at the conclusion that the specific 
heat at constant volume increases with the temperature, and at 
4400° C. attains the value 9*6. M. Berthelot's experiments 
do not, however, agree with those of MM. Le Chatelier and 
Mallard, and two series of experiments conducted by the latter 
experimenters do not agree with one another. The specific 
heat at constant volume may, however, be calculated from 
the. velocity of explosion with the aid of the proposed formula. 
A few explosions have therefore been selected and the specific 
heats and temperatures calculated from them ; specific heats 
at intermediate temperatures being found by interpolation. It 
was immediately perceived that the specific heats of O a , H 2 , 
N 2 , and CO might for all practical purposes be taken as 
identical at all temperatures. 

A few words are necessary regarding explosions in which 
water is formed. If the specific heat of steam is taken as 
f x specific heat of the diatomic gases, the found rates of ex- 
plosion fall below the calculated rates when the dilution with 
inert gas is great, and vice versa when the dilution is small. 
It is possible to account for this by two theories. The first 
theory is that at high temperatures the water is dissociated, 
whereas at low temperatures the combination of hydrogen and 
oxygen is complete. The second theory is that the specific heat 

H 2 


Mr. D. L. Chapman on the 





















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Rate of Explosion in Gases. 


of steam rises more rapidly with the temperature than the spe- 
cific heat of the diatomic gas. The theory of dissociation is 
rendered improbable by the fact that dilution of electrolytic 
gas with oxygen lowers the rates a little more than dilution 
with an equal volume of nitrogen. The adoption of such an 
hypothesis would render it necessary for us to suppose that the 
chemical reaction does not proceed to its limit. Moreover, it 
would make it difficult to calculate the rates whenever steam 
is formed, for it would then be impossible, with our present 
knowledge, to say how far the chemical reaction would pro- 
ceed in any particular case. We are therefore encouraged to 
test the first theory, i. e. that the specific heat of steam rises 
more rapidly with the temperature than that of the diatomic 
gases. The specific heat of steam at different temperatures 
has therefore been calculated from a few selected rates, as in 
the case of the elementary gases; and the values thus found 
are used to calculate the other rates. The results are given 
below. (Table II.) 

Table II. — Specific Heats at Different Temperatures. 
w = specific heat of water, g = specific heat of diatomic gases. 





























































































Mr. D. L. Chapman on the 


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Rate of Explosion in Gases. 




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104 Dr. 0. Barus on the Aqueous Fusion of Glass. 

On referring to the explosion of ethylene with excess of 
oxygen it is seen that C0 2 is not completely dissociated until 
a temperature of 3500° C. is reached. In all cases the tem- 
perature of explosion of cyanogen with excess of oxygen is 
above this, and therefore C0 2 is never formed. 

The Pressure of Explosion. 
The maximum pressure of explosion may be calculated with 
the aid of the two formulae 

llY 2 

These two equations lead to the formula 
/*V 2 C, ^ 

The pressure for an explosion of equal volumes of cyanogen 
and oxygen calculated from this formula is 57 atmospheres. 
Jones and Bower "* by breaking glass tubes obtain the value 
58 atmospheres 

VII. The Aqueous Fusion of Glass, its Relation to Pressure 
and Temperature. First Paper. By Carl Barus f. 

SOME time ago I published J a series of results due to the 
action of hot water at 185° on glass, the water being- 
kept liquid by pressure. It was shown that the water con- 
tained in sealed capillary glass tubes increased in compressi- 
bility while it steadily diminished in bulk, as described in the 
subjoined summary of two consistent experiments with 
different tubes. During the observations the column soon 
became turbid, but it remained translucent enough to admit 
of measurement. As the action at 185° proceeded, the 
length of the thread of water decreased. This thread was 
contained within the walls of the tube between two terminal 
threads of mercury (the lower being movable and trans- 
mitting pressure), and therefore decrease in the length of the 
thread can only mean contraction of volume of the system of 
glass and water in contact. The results are as follows : — 
6 denoting the temperature of the capillary thread (main- 
tained constant by a transparent vapour-bath) ; t the time 

* Journal of the Manchester Lit. and Phil. Soc. 1898. 

t Communicated by the Author. 

| Barus : American Journal of Science, xli. p. 110 (1891). 

Capillary tube, with appurtenances, 
for measuring the Compressibility 
of Liquids. 

s ■ * — 

6"- - — 

AABB. Flange screwing to compression-pump. 

a be. Capillary tube containing thread of water between visible 

threads of mercury. Ends of threads at S and S'. 
G G. Water-bath to cool paraffin plug- of capillary tube. 
eeee. Annular vapour-bath of glass, containing charge, hh, of aniline 
oil, kept in ebullition by the ring-burner HE. Non- 
conducting jacketing of tube and screens not shown. 
D tubulure for condenser, T for thermometer. 
Observations made with the cathetometer observing each mercury 
meniscus along the lines of sight S, S' , through the clear walls of the glass 
vapour-bath eeee. 


Dr. C. Barus on the 

of the observation counting from the beginning of ebul- 
lition in the vapour-bath ; v the total increment of volume 
due to the thermal expansion, V the total volume, so that 
v/Y is the mean expansion per uuit of volume at 185° ; /3 the 
mean compressibility within 300 atmospheres. 

Table showing Thermal Expansion and Compressibility of 
Silicated Water at 185° and 20 to 300 atmospheres. 
Diameter of tube '045 centim. Length of column of 
water at 24°, 14 centim. 


v/V . 10 3 . 

(3 . 10 G . 



v/V . 10 3 . 

/3.10 6 . 














+ 103 




+ 27 


























At the conclusion of the experiment the thread was solid, 
as I supposed, at high pressure (300 atm.), though not 
apparently so at low pressure. This was inferred since the 
mercury thread advancing under pressure did not, on removal 
of pressure, return again as a whole, but broke into small 
parts in a way to make further measurement without 
immediate value*. On breaking the tube apart after cooling, 
and examining it under the microscope, the capillary canal 
was found to be nearly, if not quite, filled with a white glassy 
incrustation. This shows that the glass swells in marked 
degree on hydration, whereas the combined volume of glass 
and water put into action, simultaneously contracts. If the 
values of v/Y given in the table be examined, it appears that 
whereas the original volume increment per unit of volume is 
greater than *103 for the rise of temperature from 24° to 185° 
at 20 atmospheres, this increment has nearly vanished after 
50 minutes of reaction. The thread at 185° is now only as 
long as it was at 24°. After 60 minutes of reaction it is even 
markedly shorter at 185° than at 24°, pressure remaining 
constant throughout. 

At the same time the compressibility, /3, of the silicated 

* A succeeding paper will take up the research from this point 

Aqueous Fusion of Glass. 107 

water at 185° is found to increase regularly from '000077 
near the beginning of the experiment, to over '000221, or 
more than three times its initial value on the same isothermal 
(185°). This result is wholly unexpected, since without 
exception the effect of solution is a decrease of the compressi- 
bility of the solvent, in proportion as more body is dissolved. 
Silicated water in the present experiment shows the reverse 
effect. Now although the hydration increases the volume of 
the glass, the gradual choking of the capillary canal goes on 
uniformly from top to bottom of the thread of water. Hence, 
since the bore is diminished at the same rate throughout the 
wetted tube, the observations for compressibility would remain 
to the same degree unchanged, ccet. par. Supposing that fine 
particles of the glass were broken off* and gradually accumu- 
lated on the mercury meniscus near the bottom of the thread 
of water, it would be possible to account for the data for /3 
cited, in consideration of the gradual constriction of the 
thread near the bottom. In such a case, however, the com- 
pressing thread of mercury would not have advanced and 
retreated through this debris with the observed regularity. 
In general mere stoppage and clogging would have been 
noticed in duplicate experiments f. I also made correlative 
experiments with saturated solutions of zinc sulphate in water 
and naphthalene in alcohol. In both cases markedly increased 
compressibilities were possible in a turbid column, and due to 
the precipitation of part of the dissolved salt, isothermally, 
by pressure. During compression a part of the dissolved body 
is changed from the liquid to the solid state by pressure, and 
hence the apparent increase of compressibility. 

From this point of view I have endeavoured to account 
preliminarily for the observed regular increase of j3 given in 
the table, though I confess some reluctance to this explana- 
tion : I have supposed that the dissolved silicate is precipitated 
out of solution by pressure and redissolved on removing 
pressure, thus producing accentuated compressibility ; that 
this effect increases as more silicate is taken up in solution, 
until finally the whole thread becomes too viscous for further 
observation. However this may be, the fact of a regularly 
and enormously increased compressibility remains as colla- 
teral evidence of the stage of progress of the reaction. 

2. There is a final result to be obtained from this experi- 
ment, and it is to this that my remarks chiefly apply. The 
reaction of the water on the glass must be along the surface 

* This does not occur. See below. 

t Thus for instance the thickness of the thread of mercury seen in the 
cathetometer did not seem to diminish. 

108 Dr. 0. Barus on the Aqueous Fusion of Glass. 

of contact of both bodies. For a given length f thread this 
surface decreases as the radius, r, of the tube. On the other 
hand, the volume of water decreases as the square of the radius, 
r, of the capillary tube. In fact, if V be the volume and S 
the surface for a given length of thread, S/V = 2/r. Let a be 
the rate of absorption of water in glass, i. e., the number of 
cub. centim. of water absorbed per square centim. of surface 
of contact, per minute. If v is the volume absorbed by S 
cm. 2 per minute, v = a S and therefore v/V = 2 ajr. Hence, if 
r is large, the apparent effect of absorption vanishes ; but in 
proportion as r is smaller, or as the tube becomes more finely 
capillary, the effect of absorption will become more obvious 
to the eye. In other words, the length of the column of water 
included between the two terminal threads of mercury will 
decrease faster for small values of the capillary radius. 
In the above results v/V taken directly from the table is 
about *003 cub. centim. per minute. The diameter of the tube 
measured microscopically was found to be about '045 centim. 
Therefore a = '000034 cub. centim. is the volume of water 
absorbed per square centim. of surface of contact, per minute, 
at 185°. This is about 180 kg., per sq. metre, per year, at 185°. 

True the phenomenon is not quite so simple as here com- 
puted, for as the action proceeds the water holds more body 
in solution, the area of unchanged glass increases, and 
possibly the liquid must diffuse or percolate through the 
layer of opalescent accretion to reach it. As against the 
seriousness of this consideration, one may allude to the 
regularity of the above results in the lapse of time and the 
occurrence of a reaction rather accelerated as time increases. 

In view of the large surfaces of reaction available even in 
small bulks of porous or triturated rock and the fact that the 
intensity of the reaction increases rapidly with temperature, 
I cannot but regard this result as important. Direct experi- 
ments * have been made with care to detect a possible thermal 
effect (rise of temperature) of the action of water on hot glass, 
but thus far without positive results. The difficulties of such 
experiments are very great. To insure chemical reaction, 
they must be made with superheated water under pressure, 
with allowances for heat conduction &c, all of wdiich make the 
measurement of small increments of temperature very 
uncertain. If, however, rise of temperature may be associated 
with the marked contraction of volume in the system water- 
glass specified, one may note, in the first place, that for a 

* G. F. Becker : Monographs U. S. Geolog. Survey, No. III., 1882. 
I have since made similar experiments with superheated water (200°). 

Uranium Radiation and Electrical Conduction produced. 109 

capillary canal about one half millimetre in diameter, the 
absorption of water is as great as 18 per cent, of the volume 
contained per hour. In finely porous rock correspondingly 
larger absorptions are to be anticipated. Again, the tempera- 
tures and pressures given in the above experiments would be 
more than reached by a column of water penetrating a few 
miles below the earth's surface. Finally, the action of water 
on silicates will be accelerated in proportion as higher 
temperatures are entered with increasing terrestrial depth. 
Eventually, therefore, heat must be evolved more rapidly than 
it is conducted away. 

With the above proviso, one may reasonably conclude that 
the action of hot water on rock within the earth constitutes a 
furnace whose efficiency increases in marked degree with the 
depth of the seat of reaction below sea-level. 

Brown University, 
Providence, U.S.A. 

VIII. Uranium Radiation and the Electrical Conduction pro- 
duced by it. By E. Butherford, M.A., B.Sc, formerly 
1851 Science Scholar, Coutts Trotter Student, Trinity 
College, Cambridge ; McDonald Professor of Physics, 
McGill University, Montreal*. 

THE remarkable radiation emitted by uranium and its 
compounds has been studied by its discoverer, Becquerel, 
and the results of his investigations on the nature and pro- 
perties of the radiation have been given in a series of papers 
in the Comptes Rendus^. He showed that the radiation, con- 
tinuously emitted from uranium compounds, has the power 
of passing through considerable thicknesses of metals and 
other opaque substances ; it has the power of acting on a 
photographic plate and of discharging positive and negative 
electrification to an equal degree. The gas through which 
the radiation passes is made a temporary conductor of electri- 
city and preserves its power of discharging electrification for 
a short time after the source of radiation has been removed. 

The results of Becquerel showed that Bontgen and uranium 
radiations were very similar in their power of penetrating 
solid bodies and producing conduction in a gas exposed to 
them ; but there was an essential difference between the two 
types of radiation. He found that uranium radiation could 
be refracted and polarized, while no definite results showing 


* Communicated by Prof. J. J. Thomson, F.K.S. 

t C. R. 1896, pp. 420, 501, 559, 689, 762, 1086 ; 1897, pp. 43S, 800. 

110 Prof. E. Rutherford on Uranium Radiation and 

polarization or refraction have been obtained for Rontgen 

It is the object of the present paper to investigate in more 
detail the nature of uranium radiation and the electrical 
conduction produced. As most of the results obtained have 
been interpreted on the ionization-theory of gases which was 
introduced to explain the electrical conduction produced by 
Rontgen radiation, a brief account is given of the theory and 
the results to which it leads. 

In the course of the investigation, the following subjects 
have been considered: — 

§ 1. Comparison of methods of investigation. 

§ 2. Refraction and polarization of uranium radiation. 

§ 3. Theory of ionization of gases. 

§ 4. Complexity of uranium radiation. 

§ 5. Comparison of the radiation from uranium and its 

§ 6. Opacity of substances for the radiation. 

§ 7. Thorium radiation. 

§ 8. Absorption of radiation by gases. 

§ 9. Variation of absorption with pressure. 
§ 10. Effect of pressure of the gas on the rate of discharge. 
§ 11. The conductivity produced in gases by complete 

absorption of the radiation. 
§ 12. Variation of the rate of discharge with distance 

between the plates. 
§ 13. Rate of re-combination of the ions. 
§ 14. Velocity of the ions. 
§ 15. Fall of potential between two plates. 
§ 16. Relation between the current through the gas and 

electromotive force applied. 
§ 17. Production of charged gases by separation of the ions. 
§ 18. Discharging power of fine gauzes. 
§ 19. General remarks. 

§ 1. Comparison of Methods of Investigation. 

The properties of uranium radiation may be investigated 
by two methods, one depending on the action on a photo- 
graphic plate and the other on the discharge of electrification. 
The photographic method is very slow and tedious, and admits 
of only the roughest measurements. Two or three days' 
exposure to the radiation is generally required to produce any 
marked effect on the photographic plate. In addition, when 
we are dealing with very slight photographic action, the 

the Electrical Conduction produced by it. Ill 

fogging of the plate, during the long exposures required, by 
the vapours of substances * is liable to obscure the results. 
On the other hand the method of testing the electrical dis- 
charge caused by the radiation is much more rapid than the 
photographic method, and also admits of fairly accurate 
quantitative determinations. 

The question of polarization and refraction of the radiation 
can, however, only be tested by the photographic method. 
The electrical experiment (explained in § 2) to test refraction 
is not very satisfactory. 

§ 2. Polarization and Refraction. 

The almost identical effects produced in gases by uranium 
and Rontgen radiation (which will be described later) led me 
to consider the question whether the two types of radiation 
did not behave the same in other respects. 

In order to test this, experiments were tried to see if 
uranium radiation could be polarized or refracted. Becquerel f 
had found evidence of polarization and refraction, but in 
repeating experiments similar to those tried by him, I have 
been unable to find any evidence of either. A large number 
of photographs by the radiation have been taken under 
various conditions, but in no case have I been able to observe 
any effect on the photographic plate which showed the presence 
of polarization or refraction. 

In order to avoid fogging of the plate during the long- 
exposures required, by the vapours of substances, lead was 
employed as far as possible in the neighbourhood of the 
plate, as its effect on the film is very slight. 

A brief account will now be given of the experiments on 
refraction and polarization. 

Refraction. — A thick lead plate was taken and a long- 
narrow slit cut through it ; this was placed over a uniform 
layer of uranium oxide ; the arrangement was then equivalent 
to a line source of radiation and a slit. Thin prisms of glass, 
aluminium, and paraffin-wax were fixed at intervals on the 
lead plate with their edges just covering the slit. A photo- 
graphic plate was supported 5 mms. from the slit. The plate 
was left for a week in a dark box. On developing a dark 
line was observed on the plate. This line was not appreciably 
broadened or displaced above the prisms. Different sizes of 
slits gave equally negative results. If there was any appreci- 
able refraction we should expect the image of the slit to be 
displaced from the line of the slit. 

* Russell, Proc. Roy. Soc. 1897. f C, R. 1896, p. 559. 

112 Prof. E. Rutherford on Uranium Radiation and 

Becquerel* examined the opacity of glass for uranium 
radiation in the solid and also in a finely-powdered state by 
the method of electric leakage, and found that, if anything, 
the transparency of the glass for the radiation was greater in 
the finely-divided than in the solid state. 1 have repeated 
this experiment and obtained the same result. As Becquerel 
stated, it is difficult to reconcile this result with the presence 
of refraction. 

Polarization. — An arrangement very similar to that used 
by Becquerel was employed. A deep hole was cut in a thick 
lead plate and partly filled with uranium oxide. A small 
tourmaline covered the opening. Another small tourmaline 
was cut in two and placed on top of the first, so that in 
one half of the opening the tourmalines were crossed and in 
the other half uncrossed. The tourmalines were very good 
optically. The photographic plate was supported 1 to 3 mm. 
above the tourmalines. The plate was exposed four days, and 
on developing a black circle showed up on the plate, but in 
not one of the photographs could the slightest difference in 
the intensity be observed. Becquerel f stated that in his 
experiment the two halves were unequally darkened, and 
concluded from this result that the radiation was doubly 
refracted by tourmaline, and that the two rays were unequally 

§3. Theory of Ionization. 

To explain the conductivity of a gas exposed to Rontgen 
radiation, the theory % has been put forward that the rays in 
passing through the gas produce positively and negatively 
charged particles in the gas, and that the number produced 
per second depends on the intensity of the radiation and the 

These carriers are assumed to be so small that they will 
move with a uniform velocity through a gas under a constant 
potential gradient. The term ion was given to them from 
analogy with electrolytic conduction, but in using the term 
it is not assumed that the ion is necessarily of atomic dimen- 
sions ; it may be a multiple or submultiple of the atom. 

Suppose we have a gas between two plates exposed to the 
radiation and that the plates are kept at a constant difference 
of potential. A certain number of ions will be produced 
per second by the radiation and the number produced will in 
general depend on the pressure of the gas. Under the electric 

* C. It. 1896, p. 559. t C. R. 1896, p. 559. 

% J. J. Thomson and E. Rutherford, Phil. Mag, Nov. 1896. 

the Electrical Conduction produced by it. 113 

field the positive ions travel towards the negative plate and 
the negative ions towards the other plate, and consequently 
a current will pass through the gas. Some of the ions 
will also recombine, the rate of recombination toeing pro- 
portional to the square of the number present. The current 
passing through the gas for a given intensity of radiation will 
depend on the difference of potential between the plates, but 
when the potential-difference is greater than a certain value 
the current will reach a maximum. When this is the case 
all the ions are removed by the electric field before they 
can recombine. 

The positive and negative ions will be partially separated 
by the electric field, and an excess of ions of one sign may be 
blown away, so that a charged gas will be obtained. If the ions 
are not uniformly distributed between the plates, the potential 
gradient will be disturbed by the movement of the ions. 

If energy is absorbed in producing ions, we should expect 
the absorption to be proportional to the number of ions pro- 
duced and thus depend on the pressure. If this theory be 
applied to uranium radiation we should expect to obtain the 
following results : — 

(1) Charged carriers produced through the volume of the 

(2) Ionization proportional to the intensity of the radiation 

and the pressure. 

(3) Absorption of radiation proportional to pressure. 

(4) Existence of saturation current. 

{>)) Rate of recombination of the ions proportional to the 
square of the number present. 

(6) Partial separation of positive and negative ions. 

(7) Disturbance of potential gradient under certain con- 

ditions between two plates exposed to the radiation. 

The experiments now to be described sufficiently indicate 
that the theory does form a satisfactory explanation of the 
electrical conductivity produced by uranium radiation. 

In all experiments to follow, the results are independent of 
the sign of the charged plate, unless the contrary is expressly 

§ 4. Complex Nature of Uranium Radiation. 

Before entering on the general phenomena of the conduction 
produced by uranium radiation, an account will be given of some 
experiments to decide whether the same radiation is emitted 
by uranium and its compounds and whether the radiation is 

Phil. Mag. S. 5. Vol. 47. No. 284. Jan. 1899. I 

114 Prof. E. Rutherford on Uranium Radiation and 

homogeneous. Rontgen and others have observed that the 
tf-rays are in general of a complex nature, including rays of 
wide differences in their power of penetrating solid bodies. 
The penetrating power is also dependent to a large extent on 
the stage of exhaustion of the Crookes tube. 

In order to test the complexity of the radiation, an electrical 
method was employed. The general arrangement is shown 
in fig. 1. 

Ffc. 1. 

The metallic uranium or compound of uranium to be 
employed was pow T dered and spread uniformly over the centre 
of a horizontal zinc plate A, 20 cm. square. A zinc plate B, 
20 cm. square, w y as fixed parallel to A and 4 cm. from it. 
Both plates were insulated. A was connected to one pole of 
a battery of 50 volts, the other pole of which was to earth ; 
B was connected to one pair of quadrants of an electrometer, 
the other pair of which was connected to earth. 

Under the influence of the uranium radiation there was a 
rate of leak between the two plates A and B. The rate of 
movement of the electrometer-needle, when the motion was 
steady, was taken as a measure of the current through the 

Successive layers of thin metal foil were then placed over 
the uranium compound and the rate of leak determined for 
each additional sheet. The table (p. 115) shows the results 
obtained for thin Dutch metal. 

In the third column the ratio of the rates of leak for each 
additional thickness of metal leaf is given. Where two 
thicknesses were added at once, the square root of the observed 
ratio is taken, for three thicknesses the cube root. The 
table shows that for the first ten thicknesses of metal the rate 
of leak diminished approximately in a geometrical progression 
as the thickness of the metal increased in arithmetical pro- 

the Electrical Conduction produced by it, 

Thickness of Metal Leaf -00008 cm. 
Layer of Uranium Oxide on plate. 


Number of 

Leak per nrin. in 

Ratio for each 
































It will be shown later (§ 8) that the rate of leak between 
two plates for a saturating voltage is proportional to the 
intensity of the radiation after passing through the metal. 
The voltage of 50 employed was not sufficient to saturate the 
gas, but it was found that the comparative rates of leak under 
similar conditions for 50 and 200 volts between the plates 
were nearly the same. When we are dealing with very 
small rates of leak, it is advisable to employ as small a voltage 
as possible, in order that any small changes in the voltage 
of the battery should not appreciably affect the result. For 
this reason the voltage of 50 was used, and the comparative 
rates of leak obtained are very approximately the same as 
for saturating electromotive forces. 

Since the rate of leak diminishes in a geometrical pro- 
gression with the thickness of metal, we see from the above 
statement that the intensity of the radiation falls off in a 
geometrical progression, i. e. according to an ordinary absorp- 
tion law. This shows that the part of the radiation considered 
is approximately homogeneous. 

With increase of the number of layers the absorption 
commences to diminish. This is shown more clearly by using 
uranium oxide with layers of thin aluminium leaf (see table 
p. 116). 

It will be observed that for the first three layers of aluminium 
foil, the intensity of the radiation falls on according to the 
ordinary absorption law, and that, after the fourth thickness, 
the intensity of the radiation is only slightly diminished by 
adding another eight layers. 


116 Prof. E. Kutherford on Uranium Radiation and 
Thickness of Aluminium foil *0005 cm. 

Number of Layers 

Leak per min. in. 


of Aluminium foil. 

















The aluminium foil in this case was about '0005 cm. thick, 
so that after the passage of the radiation through '002 cm. of 
aluminium the intensity of the radiation is reduced to about 
2V of its value. The addition of a thickness of *001 cm. of 
aluminium has only a small effect in cutting down the rate 
of leak. The intensity is, however, again reduced to about 
half of its value after passing through an additional thickness 
of '05 cm., which corresponds to 100 sheets of aluminium 

These experiments show that the uranium radiation is 
complex, and that there are present at least two distinct types 
of radiation — one that is very readily absorbed, which will be 
termed for convenience the a radiation, and the other of a 
more penetrative character, which will be termed the ft 

The character of the j3 radiation seems to be independent 
of the nature of the filter through which it has passed. It 
was found that radiation of the same intensity and of the same 
penetrative power was obtained by cutting off the a radiation 
by thin sheets of aluminium, tinfoil, or paper. The /S radiation 
passes through all the substances tried with far greater facility 
than the a radiation. For example, a plate of thin cover- 
glass placed over the uranium reduced the rate of leak to ^ 
of its value; the j3 radiation, however, passed through it 
with hardly any loss of intensity. 

Some experiments with different thicknesses of aluminium 
seem to show, as far as the results go, that the /3 radiation is 
of an approximately homogeneous character. The following 
table gives some of the results obtained for the ft radiation 
from uranium oxide : — 

the Electrical Conduction 'produced by it. 
fi Radiation. 


Thickness of 

Rate of Leak. 




The rate of leak is taken as unity after the a radiation has 
been absorbed by passing through ten layers of aluminium 
foil. The intensity of the radiation diminishes with the 
thickness of meted traversed according to the ordinary 
absorption Jaw. It must be remembered that when we are 
dealing with the ft radiation alone, the rate of leak is in 
general only a few per cent, of the leak due to the a radiation, 
so that the investigation of the homogeneity of the /3 radiation 
cannot be carried out with the same accuracy as for the a, 
radiation. As far, however, as the experiments have gone, 
the results seem to point to the conclusion that the /3 radiation 
is approximately homogeneous, although it is possible that 
other types of radiation of either small intensity or very 
great penetrating power may be present. 

§ 5. Radiation emitted by different Compounds of Uranium. 

All the compounds of uranium examined gave out the two 
types of radiation, and the penetrating power of the radiation 
for both the a and /3 radiations is the same for all the compounds. 

The table (p. 118) shows the results obtained for some of 
the uranium compounds. 

Fig. 2 shows graphically some of the results obtained for 
the various uranium compounds. The ordinates represent 
rates of leak, and the abscissas thicknesses of aluminium 
through which the radiation has passed. 

The different compounds of uranium gave different rates 
of leak, but, for convenience of comparison, the rate of leak 
due to the uncovered salt is taken as unity. 

It will be seen that the rate of decrease is approximately 
the same for the first layer of metal, and that the rate of 
decrease becomes much slower after four thicknesses of foil. 

The rate of leak due to the /3 radiation is a different 
proportion of the total amount in each case. The uranium 

118 Prof. E. Rutherford on Uranium Radiation and 
Thickness of Aluminium foil *0005 cm. 

Number of 

Layers of 

Aluminium foil. 

Proportionate Rate of Leak. 




















Fig. 2. 








X 5 


V \ 








;m r'orfi 

ss/cw Si 

1 0X/D£ 


10 E0 30 40 

Thickness of Aluminium : each division -00012 cm. 

the Electrical Conduction produced by it. 


metal was used in the form of powder, and a smaller area of 
it was used than in the other cases. For the experiments on 
uranium oxide a thin layer of fine powder was employed, 
and we see, in that case, that the ft radiation bears a much 
smaller proportion to the total than for the other compounds. 
When a thick layer of the oxide was used there was, how- 
ever, an increase in the ratio, as the following table shows : — 

Number of layers 
of Aluminium foil. 

Rate of Leak. 

Thin layer of 
Uranium Oxide. 

Thick layer of 
Uranium Oxide. 



















The amount of the a radiation depends chiefly on the 
surface of the uranium compound, while the ft radiation 
depends also on the thickness of the layer. The increase of 
the rate of leak due to the ft radiation with the thickness of 
the layer indicates that the ft radiation can pass through a 
considerable thickness of the uranium compound. Experi- 
ments showed that the leak due to the a radiation did not 
increase much with the thickness of the layer. I did not, 
however, have enough uranium salt to test the variation of 
the rate of leak due to the ft radiation for thick layers. 

The rate of leak from a given weight of uranium or 
uranium compound depends largely on the amount of surface. 
The greater the surface, the greater the rate of leak. A small 
crystal of uranium nitrate was dissolved in water, and the 
water then evaporated so as to deposit a thin layer of the 
salt over the bottom of the dish. This gave quite a large 
leakage. The leakage in such a case is due chiefly to the 
a radiation. 

Since the rate of leak due to any uranium compound 
depends largely on its amount of surface, it is difficult to 
compare the quantity of radiation given out by equal amounts 
of different salts : for the result will depend greatly on the 

120 Prof. E. Rutherford on Uranium Radiation and 

state of division of the compound, It is possible that the 
apparently very powerful radiation obtained from pitchblende 
by Curie * may be partly due to the very fine state of division 
of the substance rather than to the presence of a new and 
powerful radiating substance. 

The rate of leak due to the /3 radiation is, as a rule, small 
compared with that produced by the a radiation. It is 
difficult, however, to compare the relative intensities of the 
two kinds. The a radiation is strongly absorbed by gases 
(§8), while the /3 radiation is only slightly so. It will be. 
shown later (§ 8) that the absorption of the radiation by the 
gas is approximately proportional to the number of ions 
produced. . If therefore the j3 radiation is only slightly 
absorbed by the gas, the number of ions produced by it is 
small, i. e. the rate of leak is small. The comparative rates 
of leak due to the a and /3 radiation is thus dependent on the 
relative absorption of the radiations by the gas as well as on 
the relative intensity. 

The photographic actions of the a and /3 radiations have 
also been compared. A thin uniform layer of uranium oxide 
was sprinkled over a glass plate ; one half of the plate was 
covered by a piece of aluminium of sufficient thickness to prac- 
tically absorb the a radiation. The photographic plate was fixed 
about 4 mm. from the uranium surface. The plate was exposed 
48 hours, and, on developing, it was found that the darkening of 
the two halves w T as not greatly different. On the one half of the 
plate the action was due to the j3 radiation alone, and on the 
other due to the a\ and ft radiations together. Except when 
the photographic plate is close to the uranium surface, the 
photographic action is due principally to the j3 radiation. 

§ 6. Transparency of Substances to the two Types of 

If the intensity of the radiation in traversing a substance 
diminishes according to the ordinary absorption law, the 
ratio r of the intensity of the radiation after passing through 
a distance d of the substance to the intensity when the 
substance is removed is given by 

r = e , 

where X is the coefficient of absorption and e=2*7. 

In the following table a few values of X are given for the 
a and @ radiations, assuming in each case that the radiation is 

* C. R. July 1898, p. 175. 

the Electrical Conduction produced by it. 


simple and that the intensity falls off according to the above 
law : — 



for the a radiation. 


for the /3 radiation. 

Dutch metal 











The above results show what a great difference there is in 
the power of penetration of the two types of radiation. The 
transparency of aluminium for the (3 radiation is over 100 
times as great as for the a radiation. The opacity of the 
metals aluminium, copper, silver, platinum for the /3 radiation 
follows the same order as their atomic weights. Aluminium 
is the most transparent of the metals used, but glass is more 
transparent than aluminium for the (3 radiation. Platinum 
has an opacity 16 times as great as aluminium. For the 
a radiation, aluminium is more transparent than Dutch metal 
or tinfoil. 

For a thickness of aluminium "09 cm. the intensity of the 
ft radiation was reduced to '25 of its value ; for a thickness 
of copper '03 cm. the intensity was reduced to *23 of its 
value. These results are not in agreement with some given 
by Becquerel * 9 who found copper was more transparent than 
aluminium for uranium radiation. 

The /3 radiation has a penetrating power of about the same 
order as the radiation given out by an average #-ray bulb. 
Its power of penetration is, however, much less than for the 
rays from a " hard " bulb. The a radiation, on the other 
hand, is far more easily absorbed than rays from an ordinary 
bulb, but is very similar in its penetrating power to the 
secondary radiation | sent out when .z-rays fall upon a metal 

It is possible that the a radiation is a secondary radiation 
set up at the surface of the uranium by the passage of the 
/3 radiation through the uranium, in exactly the same way 

* C. R. 1896, p. 763. 

f Perrin, C. R. cxxiv. p. 455 ; Sagnac, C. R. 1898. 

122 Prof. E. Rutherford on Uranium Radiation and 

as a diffuse radiation is produced at the surface of a metal by 
the passage of Rontgen-rays through it. There is not, how- 
ever, sufficient evidence at present to decide the question. 

§ 7. Thorium Radiation. 

While the experiments on the complex nature of uranium 
radiation were in progress, the discovery * that thorium 
and its salts also emitted a radiation, which had general 
properties similar to uranium radiation, was announced. A 
few experiments were made to compare the types of radiation 
emitted by uranium and thorium. 

The nitrate and the sulphate of thorium were used and gave 
similar results, although the nitrate appeared to be the more 
active of the two. The leakage effects due to these salts were 
of quite the same order as those obtained for the uranium 
compounds; but no satisfactory quantitative comparison can 
be made between the uranium and thorium salts as the amount 
of leak depends on the amount of surface and thickness of 
the layer. 

It was found that thorium nitrate when first exposed to the 
air on a platinum plate was not a steady source of radiation, 
and for a time the rate of leak varied very capriciously, being 
sometimes five times as great as at others. The salt was very 
deliquescent, but after exposure of some hours to the atmo- 
sphere the rate of leak became more constant and allowed of 
rough comparative measurements. Thorium sulphate was 
more constant than the nitrate. 

The absorption of the thorium radiation was tested in the 
same way as for uranium radiation. The following table 
gives some of the results. The aluminium foil was of the 
same thickness ('0005 cm.) as that used in the uranium 
experiments : — ■ 

Number of Layers 
of Aluminium foil. 

Leak per minute in 










* G. C. Schmidt, Wied. Annal. May 1898. 

the Electrical Conduction produced by it. 123 

The curve showing the relation between the rate of leak 
and the thickness of the metal traversed is shown in fig. 2 
(p. 118), together with the results for uranium. 

It will be seen that thorium radiation is different in pene- 
trative power from the a radiation of uranium. The radiation 
will pass through between three and four thicknesses of alu- 
minium foil before the intensity is reduced to one-half, while 
with uranium radiation the intensity is reduced to less than a 
half after passing through one thickness of foil. 

With a thick layer of thorium nitrate it was found that the 
radiation was not homogeneous, but rays of a more penetrative 
kind were present. On account of the inconstancy of thorium 
nitrate as a source of radiation, no accurate experiments have 
been made on this point. 

The radiations from thorium and uranium are thus both 
complex, and as regards the a type of radiation are different 
in penetrating power from each other. 

In all the experiments on uranium and thorium, care was 
taken that no stray radiation was present which would obscure 
the results. Such precautions are very necessary when the 
rate of leak, due to the radiation transmitted through a con- 
siderable thickness of metal, is only a small percentage of the 
total. The method generally employed was to cover the layer 
of active salt with the metal screen, and then place in position 
over it a large sheet of lead with a rectangular hole cut in it 
of smaller area than that of the layer of salt. The lead was 
pressed tightly down, and the only radiation between the 
parallel plates had to pass through the metal screen, as the 
lead was too thick to allow any to go through. 

§8. Absorption of Uranium Radiation by Gases. 

The a. radiation from uranium and its compounds is rapidly 
absorbed in its passage through gases. The absorption for 
hydrogen, air, and carbonic acid was determined, and was 
found to be least in hydrogen and greatest in carbonic acid. 
To show the presence of absorption, the following arrange- 
ment (fig. 3) was used : — 

A layer of uranium-potassium sulphate or uranium oxide 
was spread uniformly over a metal plate P, forming a lamella 
of 11 cm. diameter. A glass vessel Gr, 12 cm. in diameter, was 
placed over the layer. Two parallel metal plates A and B, 
1*5 cm. apart, were insulated from each other by ebonite 
rods. A circular opening 7 cm. in diameter was cut in the 
plate A, and the opening covered by a sheet of aluminium 
foil O0005 cm. thick. The plate B was connected through a 

124 Prof. E. Rutherford on Uranium Radiation and 

rod R to a screw adjustment, S, so that the condenser AB 
could be moved as a whole parallel to the base-plate. The 
system AB was adjusted parallel to the uranium surface and 
did not rotate with the screw. The rod R passed through 

<*£"*> x),| h 



a short glass tube fixed in the ebonite plate C. A short 
piece of indiarubber tubing T was passed over the glass 
tube and a projecting flange in which the rod R was screwed. 
This served the same purpose as the usual stuffing-box, and 
allowed the distance of AB from the uranium to be adjusted 
under low pressures. 

The plate A was connected to one pole of a battery of 
60 volts, the other pole of which was to earth. The plate B 
was connected through the screw to one pair of quadrants of 
an electrometer, the other pair of which was to earth. In 
order to avoid the collection of an electrostatic charge on the 
glass surface due to the conduction between the uranium and 
the glass near it, it was found very necessary to coat the 
inside of the glass cylinder with tinfoil. The tinfoil and 
base-plate P were connected to earth. 

Since the surface of the uranium layer may be supposed 
to be giving out radiation uniformly from all parts, the 
intensity of the radiation at points near the centre of the 
uranium surface should be approximately uniform. If there 
were no absorption of the radiation in the gas, we should 
expect the intensity of the radiation to vary but slightly with 

the Electrical Conduction produced by it. 


distances from the surface sniull compared with the diameter 
of the radiating surface. 

The radiation passing through the aluminium produces 
conductivity between A and B (fig. 3), and the rate of 
leak depends on the intensity of the radiation which has 
passed through a certain thickness of gas and the aluminium 
foil. As the system AB is moved from the base-plate, if 
there is a rapid absorption of the radiation in the gas, we 
should expect the rate of leak to fall off rapidly, and this 
is found to be the case. The following table gives the results 
obtained for air, hydrogen, carbonic acid, and coal-gas. For 
the first reading the distance d of the aluminium foil from the 
base-plate was about 3*5 mm. 

Kate of leak between plate: 

Distance of Al. foil 
from Uranium. 





d + 1-25 mm. 
., + 2-5 „ 
„ + 3-75 „ 
, + 5 

,. + 7-5 „ 
„ +10 
„ +125 „ 
„ +15 








The rate of leak for the distance d is taken as unity in 
each gas for the purpose of comparison. The actual rates of 
leak between A and B for the distance d is given in the 
following table : — 


JRate of leak 

in scale-divisions 

per min. 







Carbonic acid 


126.. Prof. E. Rutherford on- Uranium Radiation and 

The results of the previous table are shown graphically in 
fig. 4, where the ordinates represent currents and. the abscissae 

Fisr. 4. 

distances from the base-plate. It will be seen that the 
current decreases most rapidly in carbonic acid and least in 
hydrogen. As the distance from the base-plate increases in 
arithmetical progression, the rate of leak diminishes approxi- 
mately in geometrical progression. The rapid decrease of 
the current is due to the absorption of the radiation in its 
passage through the gas. The decrease of the current in air 
at 190 mm. pressure is also shown in the figure. Since the 
absorption is smaller for air at this pressure than at normal 
pressure, the rate of leak diminishes much more slowly with 
the distance. 

In the above experiments both the a and ft radiations 
produce conductivity in the gas. A thin layer of uranium 
oxide was, however, nsed, and in that case the rate of leak due 
to the j3 radiation may be neglected in comparison with that 
produced by the a radiation. 

The results that have been obtained on the variation of the 
rate of leak with distance may be simply interpreted on the 
theory of the ionization of the gas through which the radia- 
tion passes. It is assumed that the rate of ionization is 

the Electrical Conduction produced by it. 127 

proportional to the intensity of the radiation (as is the case 
in Rontgen-ray conduction), and that the intensity of the 
radiation near the uranium surface is constant over a plane 
parallel to that surface. This is very approximately the case 
if the distance from the uranium surface is small compared 
with the diameter of the radiating surface. 

For simplicity we will consider the case of an infinite 
plane of uranium giving out homogeneous radiation. 

If I be the intensity of the radiation close to the uranium 
surface, the intensity at a distance x is equal to le~ kx where \ 
is the coefficient of absorption of the gas. The intensity is 
diminished in passing through the layer of aluminium foil A 
(fig. 3) in a constant ratio for all distances from the uranium. 
The intensity at a distance x after passing through the 
aluminium is thus Kle~ Kx where a: is a constant. The rate of 
production of the ions between two parallel planes between 
A and B (fig. 3) at distances x + dx and x from the uranium 
is therefore proportional to /cle~ xx dx. If r be the distance of 
A from the uranium, and / the distance between A and B, 
the total number of ions produced per second between A and 
B is proportional to 


I/ce~ kx dx, 


or to 


When a " saturating " electromotive force (see § 16) acts 
between A and B, the current is proportional to the total 
number of ions produced. Now, as the system AB is moved 


from the radiating surface, — (1 — e~ xl ) is a constant for any 

particular gas. We thus see that the rate of leak is propor- 
tional to e~ kr , or the rate of leak decreases in geometrical 
progression as the distance r increases in arithmetical pro- 

This result allows us to at once deduce the value of the 
coefficient of absorption for different gases from the data we 
have previously given. 

The results are given in the following table : — 

128 Prof. E. Rutherford on Uranium Radiation and 


Value of X. 



Carbonic acid 


or, to express the same results in a different way, the 
intensity of the radiation from an infinite plane of uranium 
is reduced by absorption to half its value after having passed 

3 mm. of carbonic acid, 

4' 3 mm. of air ; 

7" 5 mm. of coal-gas, 
16'3 mm. of hydrogen. 

We see that the absorption is least in hydrogen and greatest 
in carbonic acid, and follows the same order as the density of 
the gases. 

The values given above are for the <x radiation. The fi 
radiation is not nearly so rapidly absorbed as the a, but, on 
account of the small electrical leakage produced in its passage 
through the gas, it was not found feasible to measure the 
absorption in air or other gases ; 

The absorption of the a radiation by gases is very much 
greater than the absorption of rays from an ordinary Crookes' 
tube. In a previous paper * it has been shown that the value 
of X for the radiation from the particular bulb used was "01. 
The absorption coefficient for the u radiation is 1'6, or 160 
times as creat. The absorption of the radiation in gases is 
probably of the same order as the absorption for ordinary 

§ 9. Variation of Absorption with Pressure. 

The absorption of the a radiation increases with increase 
of pressure and very approximately varies directly as the 

The same apparatus was used as in fig. 3, and the vessel 
was kept connected to an air-pump. The variation of the 

* Phil. Mag, April 1897. 

the Electrical Conduction produced by it. 


rate of leak between A and B for different distances from the 
base-plate was determined for pressures of 760, 370, and 
190 mm., and the results are given below : — ■ 

Rate of leak between plates. 

Distance of A 
from Uranium. 

Air 760 mm. 

Air 370 mm. 

Air 190 mm. 

cl {— 35 mm.) 
,, + 2'5 mm. 

„ + 5 „ 
„ + 7-5 „ 
„ +10 „ 
„ +12-5 „ 
„ +15 „ 









For the purpose of comparison the rate of leak at the 
distance d is taken as unity in each case. It can readily be 
deduced from the results that the intensity of the radiation is 
reduced to half its value after passing through 

4'3 mm. c 

>f air at 760 mm. 

10 „ 

,. 370 „ 

19-5 „ 

„ 190 „ 

The absorption is thus approximately proportional to the 
pressure for the range that has been tried. It was not 
found feasible to measure the absorption at lower pressures 
on account of the large distances through which the radiation 
must pass to be appreciably absorbed, 

A second method of measuring the absorption of the radia- 
tion in gases, which depends on the variation of the rate 
of leak between two plates as the distance between them is 
varied, is given in § 12. 

§ 10. Effect of Pressure on the Rate of Discharge. 

Becquerel * has given a few results for the effects of 
pressure, and showed that the rate of leak due to uranium 
diminished with the pressure. Beattie and S. de Sinolan f also 

* Comptes Rendzis, p. 438 (1897). 
t Phil. Mag. xliii. p. 418 (1897). 

Phil. Mag, 8. 5. Vol. 47. No. 284. Jan. 1*99. 


130 Prof. E. Rutherford on Uranium Radiation and 

investigated the subject, and came to the conclusion that in 
some cases the rate of leak varied as the pressure, and in 
others as the square root of the pressure, according to the 
voltage employed. Their tabulated results, however, do not 
show any close agreement with either law, and in fact, as I 
hope to show later, the relation between the rate of leak and 
the pressure is a very variable one, depending to a large 
extent on the distances between the uranium and the sur- 

Fig. 5. 

rounding conductors, as well as on the gas employed. The 
subject is greatly complicated by the rapid absorption of the 
radiation by gases, but all the results obtained may be inter- 
preted on the assumption that the rate of production of ions 
at any point varies directly as the intensity of the radiation 
and the pressure of the gas. 

To determine the effects of pressure, an apparatus similar 
to fig. 3 was used, with the difference that the plate _ A was 
removed. The uranium compound was spread uniformly 
over the central part of the lower plate. The movable 
plate ? which was connected with the electrometer, was 10 cm, 

the Electrical Conduction produced by it. 


in diameter and moved parallel to the uranium surface. 
The base-plate was connected to one pole of a battery of 
100 volts, the other pole of which was connected to earth. 
The rate of movement of the electrometer-needle was taken 
as a measure of the current between the plates. In some 
cases the uranium compound was covered with a thin layer of 
aluminium foil, but although this diminished the rate of leak 
the general relations obtained were unaltered. 

The following tables give the results obtained for air, 
hydrogen, and carbonic acid at different pressures with a 
potential-difference of 100 volts between the plates — an 
amount sufficient to approximately " saturate " the gases air 
and hydrogen. Much larger voltages are required to produce 
approximate saturation for carbonic acid. 

Air : Uranium oxide on base-plate. Plates about 3*5 mm. 






















For hydrogen and carbonic acid. Plates about 3*5 mm. 

Hydrogen. Carbonic Acid. 


























132 Prof. E. Rutherford on Uranium Radiation and 

The current at atmospheric pressure is in each case taken as 
unity for comparison, although the actual rates of leak were 
different for the three gases. Fig. 5 (p. 130) shows these 
results graphically, where the ordinates represent current 
and the abscissas pressure. The dotted line shows the position 
of the curve if the rate of leak varied directly as the pressure. 
It will be observed that for all three gases the rate of leak 
first of all increases directly as the pressure, and then 
increases more slowly as the pressure increases. The differ- 
ence is least marked in hydrogen and most marked in 
carbonic acid. In hydrogen the rate of leak is nearly pro- 
portional to the pressure. 

The relation between the rate of leak and the pressure 
depends also on the distance between the plates. The following- 
few numbers are typical of the results obtained. There was 
a potential-difference of 200 volts between the plates and the 
rate of leak is given in scale-divisions per mm. 


Rate of Leak. 

Distance between 
plates 2*5 mm. 

Distance between 
plates 15 mm. 






1,7 i 

For small distances between the plates the rate of leak is 
more nearly proportional to the pressure than for large dis- 

The differences between the results for various gases and 
for different distances receive a simple explanation if we 
consider that the intensity of the radiation falls off rapidly 
between the plates on account of the absorption in the gas. 
The tables given for the relation between current and pres- 
sure, where the distance between the plates is small, show 
that when the absorption is small, the rate of leak varies 
directly as the pressure. For small absorption the intensity 
of the radiation is approximately uniform between the plates, 
and therefore the ionization of the gas is uniform throughout 
the volume of the gas between the plates. Since under a 
saturating electromotive force the rate of leak is proportional 

the Electrical Conduction produced by it. 133 

to the total ionization, the above experiments show that the 
rate of production of the ions at any point is proportional to 
the pressure. It has been previously shown that the absorp- 
tion of the radiation is approximately proportional to the 

Let q = rnte of production of the ions near the uranium 
surface for unit pressure. 
A = coefficient of absorption of the gas for unit pres- 

The total number of ions produced between the plates, distant 
d apart, per unit area of the plate is therefore easily seen to 
be equal to 


pq \ e-P*** da?; 

or to 

U 1 - 

-p\ Q d \ 


since we have shown that the ionization and absorption are 
proportional to the pressure. If there is a saturating electro- 
motive force acting on the gas, the ratio of the rate of leak at 
the pressure p 1 to that at the pressure p 2 is equal to the ratio r 
of the total number of ions produced at the pressure p l to the 
total number at pressure p 2 and is given by 

1 — e~ p ^o d 

Now pJ^Q is the coefficient of absorption of the gas for the 
pressure p lt If the absorption is small between the plates, 
p{k d and p^d are both small and the value of r reduces to 

rts Pi 


or the rate of leak when the pressure is small is proportional 
to the pressure. 

If the absorption is large between the plates at both the 
pressures pi and p 2 , the value of r is nearly unity — i. <?. the 
rate of leak is approximately independent of the pressure. 
Experimental results on this point are shown graphically in 
fig. 7 (p. 138). _ 

For intermediate values of the absorption, the value of r 
changes more slowly than the pressure. 

With the same distance between the plates, the difference 

134 Prof. E. Rutherford on M 


Radiation and 

between the curves (fig. 5) for air and hydrogen is due to the 
greater absorption of the radiation by the air. The less the 
absorption of the gas, the more nearly is the rate of leak pro- 
portional to the pressure. For carbonic acid the rate of 
leak decreases far more slowly with the pressure than for 
hydrogen; this is due partly to the much greater value of the 
absorption in carbonic acid and partly to the fact that 
100 volts between the plates was not sufficient to saturate the 

If we take the rate of leak between two parallel plates 
some distance from the source of radiation, we obtain the 
somewhat surprising result that the rate of leak increases at 
first with diminution of pressure, although a saturating elec- 
tromotive force is applied. 

The arrangement used was very similar to that in fig. 3. 
The rate of leak was taken between the plates A and B, which 
were 2 cm. apart, and the plate A was about 1*5 cm. from 
the uranium surface. The following table gives the results 
obtained : — 



















The current at atmospheric pressure is taken as unity. 
The results are represented graphically in Rg. 6. 

The rate of leak reaches a maximum at a pressure of less 
than half an atmosphere, and then decreases, and at a pressure 
of 100 mm. the rate of leak is still greater than at atmo- 
spheric pressure. 

This result is readily explained by the great absorption of 
the radiation at atmospheric pressure and the diminution of 
absorption with pressure. 

Let d t = distance of plate A from the uranium. 

^2— V V B „ 

the Electrical Conduction produced by it. 


With the notation previously used, the total ionization between 
A and B (on the assumption that the radiating surface is 
infinite in extent) is readily seen to be equal to 

-i- < g-Mo^l £-Mo^2 > 

This is a function of the pressure, and is a maximum when 

dje-rMj —d^e~ pK ^ = 0, 
i. e. when 

The value of pX for air at 760 mm. is 1*6. 

Fig. 6. 




If ^2 = 3 cm., ^i = l, the leak is a maximum when the 
pressure is about J of an atmosphere. On account of the 
large distance of the plates from the uranium surface in 
the experimental arrangements, no comparison between 
experiment and theory could be made. 

In all the investigations on the relation between the pres- 
sure and the rate of leak, large electromotive forces have 
been used to ensure that the current through the gas is 
proportional to the total ionization of the gas. With low 
voltages the relation between current and pressure would be 

136 Prof. E. Rutherford on Uranium Radiation and 

very different, and would vary greatly with the voltage and 
distance between the electrodes as well as with the gas. It 
has not been considered necessary to introduce the results 
obtained for small voltages in this paper, as they are very 
variable under varying conditions. Although they may all be 
simply explained on the results obtained for the saturating 
electromotive forces they do not admit of simple calculation, 
and only serve to obscure the simple laws which govern the 
relations between ionization, absorption, and pressure. The 
general nature of the results for low voltages can be deduced 
from a consideration of the results given for the connexion 
(see § 16) between the current through the gas and the 
electromotive force acting on it at various pressures. 

The above results for the relation between current and 
pressure may be compared with those obtained for Rontgen 
radiation. Perrin * found that the rate of leak varied directly 
as the pressure for saturating electromotive forces when the 
radiation did not impinge on the surface of the metal plates. 
This is in agreement with the results obtained for uranium 
radiation, for Perrin's result practically asserts that the ioni- 
zation is proportional to the pressure. The results, however, 
of other experimenters on the subject are very variable and 
contradictory, due chiefly to the fact that in some cases the 
results were obtained for non-saturating electromotive forces, 
while, in addition, the surface ionization at the electrodes 
greatly complicated the relation, especially at low pressures. 

§11. Amount of Ionization in Different Gases. 

It has been shown that the oc radiation from uranium is 
rapidly absorbed by air and other gases. In consequence of 
this the total amount of ionization produced, when the radia- 
tion is completely absorbed, can be determined. 

The following arrangement was used : — A brass ball 2*2 cm. 
in diameter was covered with a thin layer of uranium oxide. 
A thin brass rod was screwed into it and the sphere was fixed 
centrally inside a bell-jar of 13 cm. diameter, the brass rod 
passing through an ebonite stopper. The bell-jar was fixed 
to a base-plate, and was made air-tight. The inside and out- 
side of the bell-jar were covered with tinfoil. In practice an 
E.M.F. of 800 volts was applied to the outside of the bell- 
jar. The sphere, through the metal rod, was connected to 
one pair of quadrants of an electrometer. It was assumed 
that, with such a large potential-difference between the bell- 
jar and the sphere, the gas was approximately saturated and 

* Comptes Rendus, cxxiii. p. 878. 

the Electrical Conduction produced by it. 


the rate of movement of the electrometer-needle was pro- 
portional to the total number of ions produced in the gas. 
The following were some of the results obtained, the rate of 
leak due to air being taken as 100. 





H vdroffen 

1 g 

Hydrochloric Acid Gas ... 

The results for hydrochloric acid and ammonia are only 
approximate, for it was found that both gases slightly altered 
the radiation emitted by the uranium oxide. For example, 
before the introduction of the gas the rate of leak due to air 
was found to be 100 divisions in 69 sec; after the introduction 
of hydrochloric acid 100 divisions in 72 sec. ; and with air 
again after the gas was removed 100 divisions in 74 sec. 

The rate of leak is greatest in coal-gas and least in hydro- 
gen, but all the gases tried show roughly the same amount 
of ionization as air. In the case considered both kinds of 
radiation emitted by uranium are producing ionization in 
the gas. By covering over the uranium oxide with a few 
layers of thin tinfoil it was found that, for the arrangement 
used, the rate of leak due to the penetrating ray was small in 
comparison with the rate of leak due to the a, radiation. 

The effect of diminution of the pressure on the rate of leak 
for air, hydrogen, and carbonic acid is shown in fig. 7, 
where the abscissae represent pressure and the ordinates rate 
of leak. In the case of air and carbonic acid it was found 
that the rate of leak slightly increased at first with diminu- 
tion of pressure. This was ascribed to the fact that even 
with 800 volts acting between the uranium and the surround- 
ing conductor the saturation for atmospheric pressure was 
not complete. It will be observed that the rate of leak in 
air remains practically constant down to a pressure of 
400 mm., and for carbonic acid down to a pressure of 
200 mm. In hydrogen, however, the change of rate of leak 
with pressure is more rapid, and shows that all the radiation 

138 Prof. E. Rutherford on Uranium Radiation and 

emitted by the uranium was not completely absorbed at 
atmospheric pressure, so that the total ionization is pro- 
bably larger than the value given in the table. 

Fig. 7. 







b c? 

v / 


/ v/ 





1 ( 

1 1 




?£ //v 

MMS . 





Assuming that there is the same energy of radiation 
emitted whatever the gas surrounding the uranium and that 
the radiation is almost completely absorbed in the gas, we 
see that there is approximately the same amount of ionization 
in all the gases for the same absorption of energy. This is a 
very interesting result, as it affords us some information on 
the subject of the relative amounts of energy required to 
produce ionization in different gases. In whatever process 
ionization may consist there is energy absorbed, and the 
energy required to produce a separation of the same quantity 
of electricity (which is carried by the ions of the gas) is 
approximately the same in all the gases tried. 

From the results we have just given, it will be seen how 
indefinite it is to speak of the conductivity of a gas produced 
by uranium radiation. The ratio of the conductivities for 
different gases will depend very largely on the distance 
apart of the electrodes between which the rate of leak is 
observed. When the distance between the electrodes (<?. g. 
two parallel plates) is small, the rate of leak is greater in 

the Electrical Conduction produced by it. 139 

carbonic acid than in air, and greater in air than in hydrogen. 
As the distance between the plates is increased, these values 
tend to approximate equality. If, however, the rate of leak is 
taken between two plates some distance from the radiating 
surface (e. g. the plates A and B in fig. 3) , the ratio of the 
rates of leak for different gases will depend on the distance 
of the plate A from the surface of the uranium. If the 
plate A is several centimetres distant from the uranium, the 
rate of leak will be greater with hydrogen than with air, and 
greater in air than in carbonic acid — the exact reverse of the 
other case. These considerations will also apply to what is 
meant by the conductivity of a gas for uranium radiation. 

In a previous paper * I found the coefficient of absorption 
of a gas for Rontgen rays to be roughly proportional to the 
conductivity of the gas. The conductivity in this case was 
measured by the rate of leak between two plates close together 
and not far from the Crookes tube. The absorption in the 
air between the bulb and the testing apparatus was small. 
If it were possible to completely absorb the Rontgen radia- 
tion in a gas and measure the resulting conductivity, the total 
current should be independent of the gas in which the radia- 
tion was absorbed. This result follows at once if the ab- 
sorption is proportional to the ionization produced for all 
gases. The results for uranium and Rontgen radiation are 
thus very similar in this respect. 

§ 12. Variation of the Current between two Plates with the 
Distance behveen them. 

The experimental arrangement adopted was similar to that 
in fig. 3 with the plate A removed. Two horizontal polished 
zinc plates 10 cm. in diameter were placed inside a bell-jar. 
The lower plate was fixed and covered with a uniform layer 
of uranium oxide, and the upper plate was movable, by 
means of a screw, parallel to the lower plate. The bell-jar 
was air-tight, and was connected with an air-pump. The 
lower plate was connected to one pole of a battery of 200 
volts, the other pole of which was earthed, and the insulated 
top plate was connected with the electrometer. The exterior 
surface of the glass was covered with tinfoil connected to 

The following table gives the results of the variation of the 
rate of leak with distance for air at pressures of 752, 376, 
and 187 mm. The results have been corrected for change of 

* Phil. Mag. April 1897. 

140 Prof. E. Rutherford on Uranium Radiation and 

the capacity of the electrometer circuit with movement of the 

Distance between 

Hate of leak in scale-divisions per min. 

752 mm. 

376 mm. 187 mm. 














65 36 








The results are shown graphically in fig. 8, where the 

Fie-. 8. 

abscissae represent distances between the plates and the ordi- 
nates rates of leak. The values given above correspond to 
saturation rates of leak ; for 200 volts between the plates is 

the Electrical Conduction produced by it. 141 

sufficient to very approximately saturate the gas even for the 
greatest distance apart of 1*5 cm. 

It will be observed that the rate of leak increases nearly 
proportionally to the distance between the plates for short 
distances, but for air at atmospheric pressure increases very 
slowly with the distance when the distances are large. 

If there were no appreciable absorption of the radiation by 
the gas, the ionization would be approximately uniform 
between the plates, provided the diameter of the uranium 
surface was large compared with the greatest distance between 
the plates. The saturation rate of leak would in that case 
vary as the distance. If there is a large absorption of the 
radiation by the gas. the ionization will be greatest near the 
uranium and will fall off rapidly with the distance. The 
saturation rate of leak will thus increase at first with the 
distance, and then tend to a constant value when the radiation 
is completely absorbed between the plates. 

The results given in the previous table allow us to deter- 
mine the absorption coefficient of air at various pressures. 
My attention was first drawn to the rapid absorption of the 
radiation by experiments of this kind. 

The number of ions produced between two parallel plates 
distant d apart is equal to 



i. e., to 


assuming the ionization and the absorption are proportional 
to the pressure. The notation is the same as that used 
in § 10. 

For the pressure p the saturation rate of leak between the 
plates is thus proportional to 1 — e~ pk ° d . 

If p and d are varied so that p x d is a constant, the rate 
of leak should be a constant. This is approximately true as 
the numbers previously given (see fig. 8) show. It must, 
however, be borne in mind that the conditions, on which the 
calculations are based, are only approximately fulfilled in 
practice, for we have assumed the uranium surface to be 
infinite in extent and that the saturation is complete. 

The variation of the rate of leak with distance agrees fairly 
closely with the theory. When p\ d is small the rate of leak 
is nearly proportional to the distance between the plates and 
the pressure of the gas. When p\ d is large the rate of leak 
varies very slowly with the distance, 

142 Prof. E. Rutherford on Uranium Radiation and 

The value of p\ can be deduced from the experi- 
mental results, so that we have here an independent method 
of determining the absorption of the radiation at different 

The lower the pressure the more uniform is the ionization 
between the plates, so that the saturation rate of leak at low 
pressures is nearly proportional to the distance between the 
plates. This is seen to be the case in fig. 8, where the curve 
for a pressure of 187 mm. is approximately a straight line. 
Similar results have been obtained for hydrogen and carbonic 

§ 13. Rate of Recombination of the Ions. 

Air that has been blown by the surface of a uranium com- 
pound has the power of discharging both positive and negative 
electrification. The following arrangement was used to find 
the duration of the after-conductivity induced by uranium 
radiation : — A sheet of thick paper was covered over with a 
thin layer of gum-arabic, and then uranium oxide or uranium 
potassium sulphate in the form of fine powder was sprinkled 
over it. After this had dried the sheet of paper was formed 
into a cylinder with the uranium layer inside. This was then 
placed in a metal tube T (fig. 9) of 4 cm. diameter. A 

Fig. 9. 


blast of air from a gasometer, after passing through a plug C 
of cotton-wool to remove dust, passed through the cylinder T 
and then down a long metal tube connected to earth. 

Insulated electrodes A and B were fixed in the metal tube. 
The electrometer could be connected to either of the elec- 
trodes A or B. In practice the quadrants of the electro- 
meter were first connected together. The electrode A or B 
and the electrometer were then charged up to a potential of 
30 volts, and the quadrants then separated. 

When the uranium was removed there was no rate of leak 
at either A or B when a rapid current of air was sent through 
the tube. On replacing the uranium cylinder and sending a 
current of air along the tube, the electrometer showed a 

the Electrical Conduction produced by it. 143 

gradual loss of charge which continued until the electrode 
was discharged. 

When the electrode A was charged to 30 volts there was 
no rate of leak of B. The rate of leak of B or A is thus 
proportional to total number of ions in the gas. The ions 
recombine in the interval taken for the air to pass between 
A and B. The rate of leak of B for a saturating voltage, 
when A is to earth, is thus less than that of A. 

For a particular experiment the rate of leak of the electrode 
A was 146 divisions per minute. When A was connected to 
earth, the saturation rate of leak of B was 100 divisions per 
minute. The distance between A and B was 44 cm., and the 
mean velocity of the current of air along the tube 70 cm. per 
second. In the time, therefore, of *63 sec. the conductivity 
of the gas has fallen to '68 of its value. 

If we assume, as in the case of Bontgenized air *, that the 
loss of conductivity is due to the recombination of the ions, 
the variation of the number with the time is given by 

dn „ 

dt=-« n > 

where n is the number of ions per c.c. and a a constant. If 
N is the number of ions at the electrode A, the number of 
ions n at B after an interval t is given by 

n N 

JSTow the saturating rates of leak at A and B are propor- 
tional to N and n, and it can readily be deduced that the 
time taken for the number of ions to recombine to half their 
number is equal to 1*3 sec. This is a much slower rate of 
recombination than with Bontgenized air near an ordinary 
Crookes tube. 

The amount of ionization by the uranium radiation is in 
general much smaller than that due to Bontgen rays, so that 
the time taken for the ions to fall to half their number is 

The phenomenon of recombination of the ions is very similar 
in both uranium and Bontgen conduction. In order to test 
whether the rate of recombination of the ions is proportional 
to the square of the number present in the gas, the following 
experiment was performed : — 

* Phil. Mag. Nov, 1897, 

144 Prof. E. Rutherford on Uranium Radiation and 

A tube A (fig. 10) was taken, 3 metres long and 5" 5 cm. 
in diameter. A cylinder D, 25 cm. long, had its interior 
surface coated with uranium oxide. This cylinder just fitted 
the large tube, and its position in the tube could be varied 
by means of strings attached to it, which passed through 
corks at the ends of the long tube. The air was forced 

•Fiff. 10. 

D __ W& 



through the tube from a gasometer, and on entering the tube 
A passed through a plug of cotton-wool, E, in order to remove 
dust from the air and to make the current of air more 
uniform over the cross-section of the tube. The air passed 
by the uranium surface and then through a gauze L into 
the testing cylinder B of 2*8 cm. diameter. An insulated 
rod C, 1*6 cm. in diameter, passed centrally through the 
cylinder B and was connected with the electrometer. The 
cylinders A and B were connected to one pole of a battery of 
32 volts, the other pole of which was to earth. 

The potential-difference of 32 volts between B and C was 
sufficient to almost completely remove all the ions from the 
gas in their passage along the cylinder. The rate of leak of 
the electrometer was thus proportional to the number of ions 
in the gas. 

The following rates of leak were obtained for different 
distances of the uranium cylinder from the gauze L 
(table, p. 145). 

The first column of the table gives the distances of the 
end of the uranium cylinder from the gauze L. d (about 
20 cm.) was the distance for which the first measurement 
was made. In the second column the time intervals taken 
for the air to pass over the various distances are given. The 
value of t corresponds to the distance d. The mean velocity 
of the current of air along .the tube was about 25 cm. per sec. 

the Electrical Conduction produced by it. 


Di stance of 

Uranium cylinder 

from L. 


Kate of leak in 


per minute. 

rates of leak. 


d+25 cm. 
cZ+50 „ 
d+100 „ 
d+200 „ 


t-\-\ sec. 
*+2 „ 
*+4 „ 
'+8 „ 


* 87 


* 87 

In the third column are given the observed rates of leak, and 
in the fourth column the calculated values. 

The values were calculated on the assumption that the rate 
of recombination of the ions was proportional to the square 
of the number present, i. e. that 


— an' 

where n is the number of ions present and a is a constant. 
The two numbers with the asterisk were used to determine 
the constants of the equation. The agreement of the other 
numbers is closer than would be expected, for in practice the 
velocity of the blast is not constant over the cross-section, 
and there is also a slight loss of conductivity of the gas due 
to the diffusion of the ions to the side of the long tube. 

It will be observed that the rate of recombination is very 
slow when a small number of ions are present in the gas, and 
that the air preserves one quarter of its conducting power 
after an interval of 8 seconds. 

§ 14. Velocity of the Ions. 

The method* adopted to determine the velocity of the 
ions in Eontgen conduction cannot be employed for uranium 
conduction. It is not practicable to measure the rate of re- 
combination of the ions between the plates on account of the 
very small after-conductivity in such a case ; and, moreover, 
the inequality of the ionization between the two plates greatly 
disturbs the electric field between the plates. 

A comparison of the velocities, under similar conditions, of 
the ions in Eontgen and uranium conduction can, however, 

Phil. Mag. S. 

* Phil. Mag. Nov. 1897. 

5. Vol. 47. No. 284. Jan. 



146 Prof. E. Butherford on Uranium Radiation and 

be readily made. The results stow that the ions in the two 
types of conduction are the same. 

In order to compare the velocities an apparatus similar to 
fig. 10 was used. The ions were blown by a charged wire A, 
and the conductivity of the gas tested immediately afterwards 
at an electrode B, which was fixed close to A. The electrode 
A was cylindrical and fixed centrally in the metal tube L, 
which was connected to earth. For convenience of calcula- 
tion it is assumed that the electric field between the cylinders 
is the same as if the cylinders were infinitely long. 

Let a, b be the radii of the electrode A and the tube L 
(internal) ; 

Let V be the potential of A (supposed positive). 

The electromotive intensity X (without regard to sign) at 
a distance r from the centre of the tube is given by 


i b 
r lofif - 

Let u l u 2 be the velocities of the positive and negative ions 
for a potential gradient of 1 volt per cm. If the velocity is 
proportional to the electric force at any point, the distance dr 
traversed by the negative ion in the time dt is given by 

dr = X.u%dt, 

log e -rdr 

or dt = — ^f . 

\u 2 

Let r 2 be the distance from the centre from which the 
negative ion can just reach the electrode in the time t taken 
for the air to pass along the electrode. 


£ = -MU^ — ; log. 




If p 2 be the ratio of the number of the negative ions that 
reach the electrode A to the total number passing by, 

Ta — a 
then p 2 = ' 


V-a 2 
p 2 {b*-a*) log 

— - (1) 

2 V 

the Electrical Conduction produced by it. 147 

Similarly the ratio p x of the number of positive ions that 
give up their charge to the external cylinder to the total 
number is given by 

lh = W7t — (2) 

In the above equations it is assumed that the current of 
air is uniform over the cross-section of the tube, and that the 
ions are uniformly distributed over the cross-section ; also, 
that the movement of the ions does not appreciably disturb 
the electric field. Since the value of t ran be calculated from 
the velocity of the current of air and the length of the elec- 
trode, the values of the velocities of the ions under unit 
potential gradient can at once be determined. 

The equation (1) shows that p 2 is proportional to V, — 
i. e. that the rate of leak of the electrode A varies directly as 
the potential of A, provided the value of V is not large 
enough to remove all the ions from the gas as it passes 
by the electrode. This was experimentally found to be the 

In the comparison of the velocities the potential V was 
adjusted to such a value that p 2 was about one half. This was 
determined by testing the rate of leak at B with a saturating 
electromotive force. The amount of recombination of the 
ions between the electrodes A and B was very small, and 
could be neglected. 

The uranium cylinder was then removed, all the other 
parts of the apparatus remaining unchanged. An aluminium 
cylinder was substituted for the uranium cylinder, and ,^-rays 
were allowed to fall on the aluminium. The bulb and 
induction-coil were placed in a metal box in order to care- 
fully screen off all electrostatic disturbances. The rays were 
only allowed to fall on the central portion of the cylinder. 
The intensity of the rays was adjusted so that, with the same 
current of air, the rate of leak was comparable with that 
produced by the uranium. It was then found that the value 
of p2 was nearly the same as for the uranium conduction. 
For example, the rate of leak of B was reduced from 38 to 
14 scale-divisions per min. by charging A to a certain small 
potential, when the air was blown by the surface of the 
uranium. When Rontgenized air was substituted, the rate of 
leak was reduced from 50 to 18 divisions per min. under the 
same conditions. The values of p 2 were '63 and *64 respec- 
tively. This agreement is closer than would be expected, 
as the bulb was not a very steady source of radiation. 

L 2 

148 Prof. E. Rutherford on Uranium Radiation and 

This result shows that the ions in Eontgen and uranium 
conduction move with the same velocity and are probably 
identical. The velocity of an ion in passing through a gas 

is proportional to — , where e is the charge carried by the ion, 

and m its mass. Unless e and m vary in the same ratio it 
follows that the charge carried by the ion in uranium and 
Eontgen conduction is the same, and also that their masses 
are equal. 

It was found that the velocity of the negative ion was 
somewhat greater than that of the positive ion. This has 
been shown to be the case for ions produced by Eontgen 
rays *. The difference of velocity between the positive and 
negative carrier is readily shown. The rate of leak of B is 
observed when charged positively and negatively. When B 
was charged positively the rate of leak measured the number 
of negative ions that escaped the electrode A, and when 
charged negatively the number of positive ions. The rate of 
leak was always found to be slightly greater when B was 
charged negatively. This is true whether A is charged 
positively or negatively, and shows that there is an excess of 
positive ions in the gas after passing by the electrode A. 

The difference of velocities of the ions can also readily 
be shown by applying an alternating electromotive force to 
the electrode A sufficient to remove a large proportion of the 
ions as the air passes by. The issuing gas is always found 
to be positively charged, showing that there is an excess of 
positive over negative ions. 

A large number of determinations of the velocities of the 
uranium ions have been made, with steady and alternating 
electromotive forces, w T hen the air passed between concentric 
cylinders or plane rectangular plates. In consequence of the 
inequality of the velocity of the current of air over the 
cross-section of the tube, and other disturbing factors which 
could not be allowed for, the determination could not be 
made with the accuracy that was desired. For an accurate 
determination, a method independent of currents of air is 
very desirable. 

§ 15. .Potential Gradient between two Plates. 

The normal potential gradient between two plates is altered 
by the movement of the ions in the electric field. 
Two methods were used to determine the potential gradient. 

* Zeleny, Phil. Mag. July 1898. 

the Electrical Conduction produced by it. 149 

In the first method a thin wire or strip was placed between 
two parallel plates one of which was covered with uranium. 
The wire was connected with the electrometer, and after being 
left some time took up the potential in the air close to the 
wire. In the second method the ordinary mercury- or 
water-dropper was employed to measure the potential at a 

For the first method two large zinc plates were taken 
and placed horizontal and parallel to one another. A layer 
of uranium oxide was spread over the lower plate. The 
bottom plate was connected to one pole of a battery of 8 volts, 
and the top plate was connected to earth. An insulated thin 
zinc strip was placed between the plates and parallel to 
them. The strip was connected with the electrometer, and 
gradually took up the potential of the point. By moving 
the strip the potential at different points between the plates 
could be determined. 

The following table is an example of the results obtained. 

Plates 4' 8 cm. apart ; 8 volts between plates. 

Distance from 
top plate. 

Potential in 


with Uranium. 

Potential in volts 












The third column is calculated on the assumption that without 
the uranium the potential falls off uniformly between the 

The method given above is not very satisfactory when the 
strip is close to the plates, as it takes up the potential of the 
point very slowly. 

The water- or mercury-dropper was more rapid in its 
action, and gave results very similar to those obtained by the 
first method. Two parallel brass plates were placed vertically 
and insulated. One plate was connected to the positive and 

150 Prof. E. Rutherford on Uranium Radiation and 

the other to the negative pole of a battery. The middle 
point of the battery was placed to earth. The water-dropper 
was connected with the electrometer. The potential at a point 
was first determined without any uranium near. One plate 
was then removed, and an exactly similar plate, covered with 
the uranium compound, substituted. The potential of the 
point was then observed again. In this way the potential at 
any point with and without the uranium could be determined. 
The curve shown in fig. 11 is an example of the potential 
gradient observed between two parallel plates 6' 6 cm. apart. 
The dotted line represents the potential gradient when the 
uranium is removed. The ordinates represent volts and the 
abscissae distances from the plate covered with the uranium 

Fio«. ll. 


+ 2 




V >S, 



\ > 

. \ 




\ \ 









'C£ B£ t 



/A/ £MS 



\ 4 







\ < 







\ \ 

\ \ 

\ \ 

\ > 






It will be observed that the potential gradient is diminished 
near the uranium and increased near the other plate. The 
point of zero potential is displaced away from the uranium. 

From curves showing the potential gradient between two 
plates, the distribution of free electrification between the plates 
can be deduced. By Taking the first differential of the curve 

we obtain -j- , the electric force at anv point, and by taking 

dx * . d 2 v 

the second differential of the curve we obtain -j— 2 , which is 

the Electrical Conduction produced by it. 151 

equal to — ±irp, where p is the volume-density of electrifica- 
tion at any point. In order to produce the disturbance of 
the electric field shown in fig. 11, there must be an excess 
of ions of one kind distributed between the plates. Such a 
result follows at once from what has been said in regard to 
the inequality of the ionization between the plates due to the 
absorption of the radiation. 

It was found that the potential gradient approached more 
and more its undisturbed value with increase of the electro- 
motive force between the plates. The displacement of the 
point of zero-potential from the uranium surface increased 
with diminution of electromotive force. For example, for 
two plates 51 mm. apart, charged to equal and opposite 
potentials, the points of zero potential were 28, 30, 33 mm. 
from the uranium when the differences of potential between 
the plates were 16, 8, and 4 volts respectively. 

When the uranium was charged positively, the point of 
zero potential was more displaced than when it was charged 
negatively. This is due to the slower velocity of the positive 

The slope of potential very close to the surface of the 
uranium has not been investigated. The deviation from the 
normal potential slope between the plates depends very 
largely on the intensity of the ionization produced in the gas. 
With very weak ionization the normal potential gradient is 
only slightly affected. 

Child * and Zeleny f have shown that the potential gradient 
between two parallel plates exposed to Rontgen rays is not 
uniform. In their cases the ionization was uniform between 
the plates, and the disturbance in the field manifested itself in 
a sudden drop at both electrodes. In the case considered 
for uranium radiation, the ionization is too small for this 
effect to be appreciable. The disturbance of the field is due 
chiefly to the inequality of the ionization, and does not only 
take place at the electrodes. 

§ 16. Relation between Current and Electromotive Force. 

The variation with electromotive force of the current 
through a gas exposed to uranium radiation has been investi- 
gated by Becquerel {, and later by de Smolan and Beattie§. 

* Wied. Annal. April 1898, p. 152, 
t Phil. Mag., July 1898. 
\ Comptes Rendus, pp. 438, 800 (1897). 
§ Phil. Mag. vol. xliii. p. 418 (1897). 

152 Prof. E. Rutherford on Uranium Radiation and 

The general relation between the current through the gas and 
the E.M.F. acting on it is very similar to that obtained for 
gases exposed to Rontgen radiation. The current at first 
increases nearly proportionally with the E.M.F. (provided 
the E.M.F/s of contact between the metals are taken into 
account), then more slowly, till finally a stage is reached, 
which may be termed the " saturation stage," where there is 
only a very slight increase of current with a very large 
increase of electromotive force. As far as experiments have 
gone, uranium oxide, when immersed in gases which do not 
attack it, gives out a constant radiation at a definite tempera- 
ture, and the variation of the intensity of radiation with the 
temperature over the ordinary atmospheric range is inap- 
preciable. For this reason it is possible to do more accurate 
work with uranium radiation than with Rontgen radiation, 
for it is almost impossible to get a really steady source of 
a?-rays for any length of time. 

It was the object of these experiments to determine the 
relation between current and electromotive force with accuracy, 
and to see whether the gas really becomes saturated ; i. €., 
whether the current appreciably increases with electromotive 
forces when the electromotive forces are great, but still not 
sufficient to break down the gas and to produce conduction 
in the gas without the uranium radiation. 

A null method was devised to measure the current, in order 
to be independent of the electrometer as a measuring instru- 
ment and to merely use it as an indicator of difference of 

Rff. 12. 



Fig. 12 shows the general arrangement of the experiment. 
A and B were two insulated parallel zinc plates : on the lower 

the Electrical Conduction produced by it. 153 

plate A was spread a uniform layer of uranium oxide. The 
bottom plate was connected to one pole of a battery of a large 
number of storage-cells, the other pole of which was to earth. 
The insulated plate B was connected to one pair of quadrants 
of an electrometer, the other pair of which was to earth. 
Under the influence of the uranium the air between the plates 
A and B is made a partial conductor, and the potential of B 
tends to become equal to that of A. In order to keep the 
potential of B at zero, B is connected through a very high 
resistance T of xylol, one end of which is kept at a steady 
potential. If the amount of electricity supplied to B through 
the xylol by the battery is equal and opposite in sign to the 
quantity passing between A and B, the potential of B will 
remain steadily at zero. In order to adjust the potential to 
be applied to one end of the xylol-tube T, a battery was 
connected through resistance-boxes Rj R 2 , the wire between 
being connected to earth. The ratio of the E.M.F. e acting 
on T to the E.M.F. E of the battery is given by 



E Bi + R 2 * 

In practice Ri + Rg was always kept constant and equal to 
10,000 ohms, and, in adjusting the resistance, plugs taken 
from one box were transferred to the other. The value of e 
is thus proportional to R 1; and the amount of current supplied 
to B (assuming xylol obeys Ohm's Law) is proportional to R x . 
If the resistances are varied till the electrometer remains at 
the " earth zero," the current between the plates is pro- 
portional to Ri. If the value of the E.M.F. applied is too 
great the needle moves in one direction, if too small in the 
opposite direction. For fairly rapid leaks the current could 
be determined to an accuracy of 1 per cent.; but for slow 
electromotive leaks this accuracy is not possible on account 
of slow changes of the electrometer zero when the quadrants 
are disconnected. 

The following tables show the results of an experiment with 
uranium oxide. The surface of the uranium was II cm. 
square. In order to get rid of stray radiation at the sides 
lead strips, which nearly reached to the top plate, were placed 
round the uranium. 16 volts were applied to the resistance- 
box, and a resistance of 10,000 ohms kept steadily in the 

154 Prof. E. Rutherford on Uranium Radiation and 

Plates 2*5 cm. apart. 

Plates *5 cm. apart. 


Current R 1 . 

Volts. 1 Current E r 














Under the column of volts the difference of potential 
between A and B is given. The current is given in terms of 
the resistance E-x required to keep the electrometer at the 
earth zero. It will be observed that for the first few readings 
Ohm's Law is approximately obeyed, and then the current 
increases more gradually till for large E.M.F/s the rate of 
increase is very slow. For the plates '5 cm. apart the 
rate of leak for 335 volts is only 50 per cent, greater than 
the rate of leak for 1 volt. 

The same general results are obtained if the surface of the 
uranium is bare or covered with thin metal. The disadvantages 
of covering the surface with thin tin or aluminium foil are (1) 
that the intensity of the radiation is considerably decreased ; 
(2) that the ions diffuse from under the tinfoil through any 
small holes or any slight openings in the side. The drawback 
of using the uncovered uranium in the form of fine powder 3 
is that under large electric forces the fine uranium particles 
are set in motion between the plates and cause an additional 
leakage. In practice the rate of leak was measured with 
potential-differences too small to produce any appreciable 
action of this kind. 

In order to investigate the current-electromotive-force 
relations for different gases the same method was used, but 
the leakage in this case took place between two concentric 
cylinders. The apparatus is shown in the lower part of 
fig. 12 : C and D were two concentric cylinders of brass 4*5 
and 3*75 cm. in diameter, insulated from each other. 

the Electrical Conduction produced by it. 


The ends of the cylinder D were closed by ebonite collars, 
and the central cylinder was supported in position by brass 
rods passing through the ebonite. The surface was uniformly 
covered with uranium oxide. The cylinder D was connected 
to one pole of a battery, the other pole of which was to earth. 
The cylinder C was connected to the electrometer. The 
following tables show the results obtained for Irydrogen, carbonic 
acid, and air. Distance between cylinders *375 cm. 


Carbonic Acid. 























































+ 1 














The above results are expressed graphically in fig. 13, 
where the ordinates represent current on an arbitrary scale 
and the abscissse volts. In the tables given for hydrogen and 
carbonic acid it will be observed that the current has a 
definite value when there is no external electromotive force 
acting. The reason for this is probably due to the contact 
difference of potential between the uranium surface and the 
interior brass surface of the outside cylinder. When the 
external cylinder was connected to earth the inside cylinder 
became charged* to —'12 volt after it was left a short time. 

* This phenomenon has been studied by Lord Kelvin, Beattie, and 
S. de Smolan, and it has been shown that metals are charged up to small 
potentials under the influence of uranium radiation. The steady 
difference of potential between two metal plates between which the 
radiation falls is the same as the contact difference of potential. An 
exactly similar phenomenon has been studied by Perrin (Comptes Rendus, 
cxxiii. p. 496) for #-rays. 

156 Prof. E. Rutherford on Uranium Radiation and 

In consequence of this action, for small electromotive forces 
the rates of leak are different for positive and negative. 

Fiar. 13. 



Acid < 

3 AS 





— ^--* 
























Results of this kind are shown more clearly in fig. 14, which 
gives the current-electromotive-force curves for hydrogen 
and carbonic acid for small voltages. When there is no 
external electromotive force acting, the current has a fixed 
value ; if the uranium is charged positively, the current 
increases slowly with the voltage ; when the uranium is 
charged negatively, the current is at first reversed, becomes 
zero, and rapidly increases with the voltage until for about 
1 volt between the plates the positive and negative currents 
are nearly equal. The curve for carbonic acid with a positive 
charge on the uranium is also shown. It will be seen that 
the initial slope of the curve is greater for carbonic acid than 
for hydrogen. 

It is remarkable that the current with zero E.M.F. for 
hydrogen is about two-thirds of its value when 216 volts are 
acting between the plates. The ions in hydrogen diffuse more 
rapidly than in air, and in consequence a large proportion 

the Electrical Conduction produced by it, 157 

of the negative ions reach the uranium and give up their 
charge to it before recombination can take place. 

Fig. 14. 

If the radiation fell between two plates of exactly the same 
metal, the inequality between the positive and negative 
current values for low voltages would almost disappear, but 
even in that case there would still be an apparent current 
through the gas, due to the fact that the negative carriers 
diffuse with greater rapidity than the positive. Effects of 
this kind have been studied for Eontgen radiation by Zeleny *. 

For large E.M.F.'s no appreciable difference in the value 
of the current could be detected whether the uranium was 
positively or negatively charged, i. e. positive and negative 
electrifications are discharged with equal facility. 

For the different gases the current tends more rapidly to a 
saturation value in hydrogen than in air, and more rapidly in 
air than in carbonic acid. In all these cases there is still a 
slight increase of current with increase of E.M.F. long after 
the " knee " of the saturation curve has been passed, and in 
no case has complete saturation been observed at atmospheric 
pressure, even for a potential gradient of 1300 volts per cm. 

The explanation of the general form of the curves showing 
the relation between current and electromotive force for 
ionized gases has been given in a previous paper f. In the 

* Phil. Mag., July 1898. 

f J. J. Thomson and E. Rutherford, Phil. Mag. November 1896. 

158 Prof. E. Rutherford on Uranium Radiation and 

case of uranium conduction the phenomenon is still further 
complicated by the want of uniformity of ionization between 
the plates and the resulting disturbance of the electrostatic 
field due to the excess of ions of one kind between the 

The ionization of the gas is greatest near the uranium sur- 
face, and falls off rapidly with the distance. The rate of 
recombination of the ions thus varies from point to point 
between the plates, being greatest near the surface of the 

The equations which express completely the relation between 
the current and electromotive force for the rate of leak between 
two parallel plates, one of which is covered with uranium, are 
very complex and cannot be expressed in simple form. The 
disturbance of the electrostatic field between the plates, due to 
the movement of the ions, has to be considered as well as 
the variable rates of recombination at the different points, and 
the difference of velocity between the positive and negative 

The great difficulty in producing complete saturation, i. e. to 
reach a stage when all the ions produced reach the electrodes, 
may be due to one or more of three causes: — 

(1) Rapid rate of recombination of the ions very near the 
surface of the uranium. 

(2) Presence of very slow moving ions together with the 
more rapidly moving carriers. 

(3) An effect of the electric field on the production of 
the ions. 

The effect of (3) is probably very small, for there is no 
experimental evidence of any such action unless the electro- 
motive forces are very high. That the slow increase of the 
current in strong fields is due to (1) rather than (2) receives 
some support from an experiment that has been recently 
tried. Instead of measuring the current with the uranium 
covering one electrode, the air which had passed over uranium 
was forced between two concentric cylinders between which 
the electromotive force was acting. The rate of leak was 
found to only increase 2 or 3 per cent, when the E.M.F. was 
increased from 16 to 320 volts. This increase is much smaller 
than in the results previously given. Since the effect of (2) 
would be present in both cases, this experiment seems to show 
that the difficulty in removing all the ions from the gas is not 
due to the presence of some very slow-moving carriers. 

the Electrical Conduction produced by it. 
Effect of Pressure. 


Some current electromotive-force curves for small voltages 
have been obtained at different pressures. Examples of the 
results are shown in fig. 15. which gives the relation between 
the current and the electromotive force at pressures of 760, 
380, 190, and 95 mm. of mercury. 

These results were obtained with a different apparatus and 
by a different method to that given in fig. 12. Two parallel 
insulated metal plates, about 3 cm. apart, one of which was 
covered with uranium oxide, were placed inside an air-tight 
vessel. One plate was connected to earth and the other to 
the electrometer. The plate connected to the electrometer 
was then charged up to a potential of 10 volts. On account 
of the presence of the uranium oxide the charge slowly leaked 
away, and the rate of movement of the electrometer-needle 
measured the current corresponding to different values of the 
electromotive force. 

Fig. 15. 





190 MM 





EN< 760 

MM. — 

* / 


Air 9 

f MM. 




The method did not admit of the accuracy of that pre- 
viously employed (see fig. 12). The rate of leak for small 
fractions of a volt could not be determined, so that in the 
curves fig. 15 it is assumed that the current was zero when 
the electromotive force was zero. This is probably not quite 

160 Prof. E. Rutherford on Uranium Radiation and 

accurate owing to the slight contact-difference of potential 
between the plates, so that there was a small initial current 
for zero external electromotive force. 

The general results show that the gas tends to become more 
readily saturated with diminution of pressure. The variation 
of the current with the E.M.F. depends on two factors — the 
velocities of the ions, and their rate of recombination. Some 
experiments on the velocity of the carriers * in ultra-violet 
light conduction showed that the velocity Of the ions in a 
given electric field is inversely proportional to the pressure. 
This is probably also true for the ions in Rontgen conduction; 
so that under the pressure of 95 mm. the ions would move 
eight times as fast as at atmospheric pressure. The variation 
of the rate of recombination with pressure has not yet been 

The curve for hydrogen at atmospheric pressure is also 
given in fig. 15, and shows that hydrogen is about as easily 
saturated as air at 190 mm. pressure. 

§ 17. Separation of the Positive and Negative Ions. 

It is a simple matter to partially separate the positive and 
negative ions in uranium conductions and produce an 
electrified gas, The subject of the production of electrifi- 
cation by passing a current of air over the surface of uranium 
enclosed in a metal vessel has been examined by Beattie f, 
who found the electrification obtained was of the same sign 
as the charge on the uranium. His results admit of a simple 
explanation on the theory of ionization. The gas near the 
surface of the uranium is far more strongly ionized than that 
some distance away on account of the rapid absorption of the 
radiation by the air. For convenience of explanation, let us 
suppose a piece of uranium, charged positively, placed inside 
a metal vessel connected to earth, and a current of air passed 
through the vessel. Under the influence of the electromotive 
force the negative ions travel in towards the uranium, and the 
positive ions towards the outer vessel. Since the ionization 
is greater near the surface of the uranium, there will be an 
excess of positive ions in the air some distance away from 
the uranium. Part of this is blown out by the current of air, 
and gives up its charge to a filter of cotton-wool. The total 
number of negative ions blown out in the same time is much 
less, as the electromotive intensity, and therefore the velocity 
of the carrier, is greater near the uranium than near the out- 
side cylinder. Consequently there is an excess of positive 

* Proc. Oamb. Phil. Soc. Feb. 21, 1888. 
f Phil. Mag. July 1897, xliv. p. 102, 

the Electrical Conduction produced by it. 161 

ions blown out, and a positively electrified gas is obtained. 
As the potential-difference between the electrodes is increased, 
the amount of electrification obtained depends on two opposing 
actions. The velocity of the carriers is increased, and conse- 
quently the ratio of the number of carriers removed is dimi- 
nished. But if the gas is not saturated, with increase of 
electromotive force the number of ions travelling between the 
electrodes is increased, and for small voltages this increase 
more than counterbalances the diminution due to increase of 
velocity. The amount of electrification obtained will there- 
fore increase at first with increase of voltage, reach a maxi- 
mum, and then diminish ; for when the gas is saturated no 
more ions can be supplied with increase of electromotive 
force. This is exactly the result which Beattie obtained, and 
which I also obtained in the case of the separation of the ions 
of Rontgenized air. The fact that more positive than nega- 
tive electrification is obtained is due to the greater velocity 
with which the negative ion travels. (See § 14.) 

The properties of this electrified gas are similar to that which 
has been found from Rontgen conduction. The opposite sign 
of the electrification obtained by Beattie for uranium in- 
duction, and by myself for Rontgen conduction *, is to be 
expected on account of the different methods employed. For 
obtaining electrification from Rontgenized air a rapid current 
of air was directed close to the charged wire. In that case 
the sign of the electrification obtained is opposite to that of 
the wire, as it is the carriers of opposite sign to the wire which 
are blown out before they reach the wire. In the case of 
uranium the current of air filled the cross-section of the space 
between the electrodes ; and it has been shown that under such 
conditions electrification of the same sign as the uranium is to 
be expected. 

§ 18. Discharging-power of Fine Gauzes. 

Air blown over the surface of uranium loses all trace of con- 
ductivity after being forced through cotton-wool or through 
any finely divided substance. In this respect it is quite 
similar to Rontgenized air. The discharging-power of cotton- 
wool and fine gauzes is at first sight surprising, for there is 
considerable evidence that the ions themselves are of molecular 
dimensions, and might therefore be expected to pass through 
small orifices ; but a little consideration shows that the ions, 
like the molecules, are continually in rapid motion, and, in 
addition, have free charges, so that whenever they approach 
within a certain distance of a solid body they tend to be 
attracted towards it, and give up their charge or adhere to 

* Phil. Mag. April 1897. 
Phil Mag. S. 5. Vol, 47. No. 284. Jan. 1899. M 

162 Uranium Radiation and Electrical Conduction. 

the surface. On account of the rapidity of diffusion * of the 
ions, the discharging-power of a metal gauze, with openings 
very large compared with the diameter of a carrier, may be 
considerable. The table below gives some results obtained for 
the discharging-power of fine copper gauze. The copper 
gauze had two strands per millim., and the area occupied by 
the metal was roughly equal to the area of the openings. The 
gauzes filled the cross-section of the tube at A (fig. 9), and 
were tightly pressed together. The conductivity of the air 
was tested after its passage through the gauzes, the velocity 
of the air along the tube being kept approximately constant. 
The rates of leak per minute due to the air after its passage 
through different numbers of gauzes is given below. 

Number of Gauzes. 

Rate of leak in divisions 
per minute. 


26 5 






After passing through 5 gauzes the conductivity of the air 
has fallen to less than ^ of its original value. Experiments 
were tried with gauzes of different degrees of coarseness with 
the same general result. The discharging-power varies with 
the coarseness of the gauze, and appears to depend more on 
the ratio of the area of metal to the area of the openings than 
on the actual size of the opening. If a copper gauze has such 
a power of removing the carriers from the gas, we can readily 
see why a small plug of cotton-wool should completely abstract 
the ions from the gas passing through it. The rapid loss of 
conductivity is thus due to the smallness of the carrier and 
the consequent rapidity of diffusion. 

§ 19. General Remarks. 

The cause and origin of the radiation continuously emitted 
by uranium and its salts still remain a mystery. All the 
results that have been obtained point to the conclusion that 
uranium gives out types of radiation which, as regards their 
effect on gases, are similar to Rontgen rays and the secondary 
radiation emitted by metals when Rontgen rays fall upon 
them. If there is no polarization or refraction the similarity 
* Townsend, Phil. Mag. June 1898. 

Notices respecting New Books. 163 

is complete. J. J. Thomson * has suggested that the re- 
grouping of the constituents of the atom may give rise to 
electrical effects such as are produced in the ionization of a 
gas. Rontgen's j and Wiedemann's J results seem to show 
that in the process of ionization a radiation is emitted which 
has similar properties to easily absorbed Rontgen radiation. 
The energy spent in producing uranium radiation is probably 
extremely small, so that the radiation could continue for long 
intervals of time without much diminution of internal energy 
of the uranium. The effect of the temperature of the uranium 
on the amount of radiation given out has been tried. An 
arrangement similar to that described in § 11 was employed. 
The radiation was completely absorbed in the gas. The 
vessel was heated up to about 200° C; but not much differ- 
ence in the rate of discharge was observed. The results of 
such experiments are very difficult to interpret, as the variation 
of ionization with temperature is not known. 

1 have been unable to observe the presence of any secondary 
radiation produced when uranium radiation falls on a metal. 
Such a radiation is probably produced, but its effects are too 
small for measurement. 

In conclusion, I desire to express my best thanks to 
Prof. J . J. Thomson for his kindlv interest and encouragement 
during the course of this investigation. 

Cavendish Laboratory, Sept. 1st, 1898. 

IX. Notices respecting New Books. 
A Treatise on Dynamics of a Particle ; ivitli numerous examples. By 

Dr. E. J. Eouth, F.R.S. Cambridge : University Press. 

Pp. xii -f 417. 
npHAT this work is a thorough one on its subject is a matter of 
-*- course, but it is more than this, it is a most interesting one. As 
Dr. South remarks in the opening words of his preface, " so many 
questions which necessarily excite our interest and curiosity are 
discussed in the dynamics of a particle that this subject has always 
been a favourite one with students." He puts the question, how- 
is it that by observing the motion of a pendulum we can tell the 
time of rotation of the earth, or, knowing this, can deduce the 
latitude of the place ? Other such problems excite our curiosity 
at the very beginning of the subject. When we study the replies 
to those problems we find new objects of interest, and so we 
mount higher and higher until we include the planetary per- 
turbations, and take account of the finite size of bodies. So far 
does Dr. Eouth carry us until he approximates quite closely to 
his familiar Rigid Dynamics. One has hitherto associated his 
work and name primarily with this latter subject, but as for 
iome forty years the whole mathematical curriculum must have 
jccupied his thoughts, he must have many potential books in his 

* Proc. Camb. Phil. Soc. vol. ix. pt. viii. p. 397 (1898). 

t Wied. Ann. lxiv. (1898). \ Zeit.f. Electrochemie, ii. p. 159 (1895). 

164 Intelligence and Miscellaneous Articles. 

mind, and we trust that his present leisure will enable him to 
collect, into book form, the accumulated stores of these past 
years. Our task is a simple one. "We shall merely indicate what 
are the subjects discussed. There are in all eight chapters. 
The headings are : — Elementary Considerations ; Rectilinear 
Motion ; Motion of Projectiles ; Constrained Motion in Two 
Dimensions ; Motion in Two Dimensions ; Central Forces ; 
Motion in Three Dimensions ; and some Special Problems. 
The work closes with two notes : the first on an Ellipsoidal 
Swarm of Particles, and the second on Lagrange's Equations, a 
new form for the Lagrangian function, and a rotating field. The 
great value to the student appears to us to be the thorough 
discussion of a large number of illustrative problems. As in his 
previous books, Dr. Routh gives ample reference to original 
memoirs, a number of historical notes, and a useful Index. Our 
reference to the Preface sufficiently indicates the wide range 
included under the heading Dynamics of a Particle. 

X. Intelligence and Miscellaneous Articles. 



To the Editors of the Philosophical Magazine. 

TN the issue of the ' Philosophical Magazine'' for December 1898, 
-■-ma note referring to the article published by me in the 
May number 1898, " On the Susceptibility of Diamagnetic and 
-Weakly Magnetic Substances," Professor Quincke draws attention 
to the fact that he had previously described* a method essentially 
similar to the one I used for the determination of the susceptibility 
and had applied it in investigating the susceptibility of Iron, 
Nickel, and Cobalt, among other things. 

That I failed to refer to this note of Professor Quincke's, and 
to a communication by Lord Kelvin t on the same subject, was due 
to the fact that I did not learn of them until after the article 
referred to above had appeared. 

I may be allowed to add that the method in the form used by 
me, involving the use of prismatic slabs transversely magnetized, 
does not admit of application to Iron, Nickel, and Cobalt, since in 
this case the induced magnetization would depend almost altogether 
upon the shape of the substance and but slightly upon its 
susceptibility^. In Lord Kelvin's note upon this subject the 
restriction that the method may be applied to those bodies only 
which are diamagnetic or slightly magnetic is implied in the title. 

Very truly yours, 
Berlin, Dec. 10, 1898. Albert P. WlLLS. 

* Tageblatt der 62 Vermmmlung der Deutsche?' Natur for seller und 
Aerzte, Heidelberg, 1889, p. 209. 

t " On a Method of determining in Absolute Measure, the Magnetic 
Susceptibility of Diamagnetic and Feebly Magnetic Solids." — Report of 
the British Association, 1890, p. 745. 

\ Maxwell, ' Electricity and Magnetism,' vol. ii. pp. 65, 66. 



MiYT.te.r-n.Bros . Coliotvpe . 

<! y 

PKil.Mag S 5.Vol.47.Pl.II . 






* fc R V *T 

[FIFTH SERIESJf f£B^° < r> 

FEBRUARY 1899. • : _,.^» 

.*'' * 

XL Radiation Phenomena in the Magnetic Field. — Magnetic 
Perturbations of the Spectral Lines. By Thomas Preston, 
M.A., D.Sc, F.R.S* 


IN the April number of this Magazine f I described a 
series of observations on u Radiation Phenomena in a *)j\ 
strong Magnetic Field. 5 '' Briefly stated, the results obtained 
showed that while the majority of spectral lines became 
triplets when the source of light was placed in the magnetic 
field and viewed across the lines of force, yet this did not 
hold good of all lines, for some were observed to be resolved 
into quartets, or sextets, or other forms by the magnetic 
field under precisely the same circumstances. 

I pointed out at that time that these quartets &c. 
might be regarded as modified forms of the normal or 
standard triplet form, and might possibly be derived from it 
by reversal. Thus, if each line of a triplet be reversed (that 
is, if an absorption-band occurs along its middle), then we 
have six lines instead of three, and so on for the other forms. 
I also pointed out, however, that the general appearance of 
these modified forms did not by any means favour that 
explanation, for they possessed none of the ordinary cha- 
racteristics of reversals. Nevertheless, this explanation could 

* Communicated by the Author. 

t Phil. Mag. vol. xlv. p. 325 (1898). The experiments described in this 
paper were performed in Oct. and Nov. 1897, and communicated to the 
Royal Dublin Society in December 1897. See Trans. Rov. Dubl. Soc. 
vol. xv. p. 385 (1898). 

Phil Mag. S. 5. Vol. 47. No, 285. Feb. 1899. N 

166 Dr. T. Preston on Radiation Phenomena 

not be ignored till it had been proved by experiment that the 
modifications were actually caused by other agencies, and 
this I endeavoured to do by two lines of attack. 

In the first place, if these phenomena are due to reversal, it 
is likely that they will cease to exist when the quantity of 
vapour in the source of light is greatly reduced. I accord- 
ingly tried sparking with weak solutions of salts instead of 
with metallic electrodes, but in no case did the quartets, or 
other modifications, reduce to the triplet form, but on the 
contrary they became clearer and more precise as the lines 
became sharper with the reduced quantity of vapour. Never- 
theless this was not regarded as conclusive or even seriously 
in opposition to the supposition of reversal, for the appearance 
of reversed lines in the strong magnetic field when the spark 
is blown about might differ from that of an ordinary reversal. 

I accordingly endeavoured to gradually increase the strength 
of the magnetic field and observe if the components of the 
supposed reversed line remained at the same distance apart 
or became more widely separated as the strength of the field 
increased. The extent to which I w 7 as able to increase the 
field at that time was not, however, sufficient to enable me to 
determine with sufficient certainty whether the reversal 
hypothesis was tenable or not. For although the com- 
ponents of the supposed reversed line appeared to separate 
under the increased field, yet this separation was not suffi- 
ciently great to overthrow the reversal hypothesis, for it 
might be said that the absorbed band along the middle of the 
line had merely become a little wider. The weight of 
evidence, however, appeared to be against the reversal theory; 
and in order to further test this matter I had a powerful 
electromagnet specially built which it was hoped would 
furnish a field sufficiently strong to determine matters 
decisively, and in this respect it has not disappointed expect- 
ation. Thanks to the courtesy of the University Authorities 
and of the Curator, Dr. W. E. Adeney *, I was able to resume 
work at the Royal University with improved apparatus, and 
it was soon found that the reversal theory must be abandoned 
and that the explanation of the various deviations from the 
normal triplet-types must be sought for in other agencies. 

Before describing these more recent results, it will render 
the explanation more intelligible if we refer for a moment to 
figs. 1, 2, 3, 4, In fig. 1 the three lines A, B, C are 

* I am deeply indebted to Dr. Adeney throughout this and the 
previous investigations, for he invited me to the Royal University 
laboratories and facilitated my work under conditions which necessarily 
interfered with his own researches. 

in the Magnetic Field. 


supposed to represent the triplet into which a spectral line of 
the standard type becomes resolved by the action of the 
magnetic field, and fig. 2 in the same way represents a 
quartet produced by the magnetic field, or, if we may say so, 

A A 

B A 

B A C 

Fi<?. 1. 

Fiff. 2. 


Fig. 3. 

B C 

Fig. 4. 

a triplet in which the middle line A has become a doublet. 
These lines are all plane-polarized, the vibrations in the side- 
lines B and C being parallel to their length, while the 
vibrations in the central constituents A are in the per- 
pendicular direction, when the light is viewed across the lines 
of force. 

Hence, if a double-image prism be placed before the slit of 
the spectroscope in the path of the beam of light, the two 
plane-polarized parts can be separated so that one part (say 
A) forms one image on the slit, while the other part (B and 
C) forms another image on the slit. These two images, being 
separated, give rise to two spectra in the field of view of the 
spectroscope which may be separated or may be caused to 
partially overlap if so desired. As a consequence a triplet or 
a quartet which appears in the field of view, as shown in 
figs. 1 and 2, without the double-image prism, becomes trans- 
formed by the double-image prism into figs. 3 and 4. Thus 
the light vibrating parallel to the lines of force (A) is 
separated from that vibrating in the perpendicular direction 
(B and C), and this facilitates observation in the case of small 
separations and in the case of overlapping lines. This sepa- 
ration of the two parts can of course be effected by a Nicol's 
prism, but the double-image prism has the advantage of 
showing the two parts simultaneously. 

The question now before us is — can the quartet shown in 
figs. 2 and 4 be derived from the triplets figs. 1 and 3 by 
mere reversal of the central line A. In answer to this 
question, it is to be remarked that it is not the central line 
A alone which shows as a doublet, for in some cases the 


168 Dr. T. Preston on Radiation Phenomena 

side lines B and C show as doublets and in others as triplets: 
and again, in some quartets the distance between the central 
pair A is almost as great as, and it may be greater than, the 
distance between the side lines B and C. Further, as the 
magnetic field increases in strength, the distance between the 
members of the central pair A (fig. 4) increases at the same 
rate as the distance between the side lines B and C. Again, 
when the side lines B and C are each resolved into doublets 
(or triplets) the separation of the constituent lines of each of 
these doublets (or triplets) increases with the magnetic field 
like the separation of the components of the normal triplet 
ABC shown in fig. 1 ; and in face of these facts the reversal 
theory becomes quite untenable. 

The general phenomenon, therefore, which remains to be 
explained is the further resolution of each constituent of the 
normal triplet into a doublet or a triplet or some other system; 
and, as we shall see immediately, the electromagnetic theory 
proposed by Dr. Larmor * may be extended to embrace all 
the phenomena yet observed. Before proceeding to consider 
this explanation, however, it is necessary to refer to a 
particular case which was recently announced as having been 
observed by MM. Becquerel and Deslandresf, and subse- 
quently by Messrs. J. S. Ames, R. F. Earhart, and H. M. 
Reese J, and which they refer to as an example of " reversed 
polarization." This phenomenon is represented to be as 

Consider the triplet shown in fig. 1 ; then in the normal 
state of affairs the vibrations in A are parallel to the lines of 
force, w r hile the vibrations in B and C are perpendicular to 
the lines of force. Now in the spectrum of iron the authors 
just named have recorded that they observed triplets in which 
the vibrations in the middle line were perpendicular to the lines 
of force, while the vibrations in the side lines were parallel to 
the lines of force — the reverse of the normal case. Stated in 
this way the phenomenon is very startling, and appears at 
first sight to be directly contradictory to all theoretical 
expectation. But if we return to fig. 4, it will be seen at 
once that this phenomenon, supposing it to exist, can be 
regarded merely as an extreme case of the quartet. For, as 
we have already said, the horizontally vibrating lines A of 
the quartet may be close together or widely separated. They 
may be even more widely separated than the vertically 
vibrating fines B and C (fig. 5), and in some particular cases 

* Dr. J. Larmor, Phil. Mag;, vol. xliv. p. 505 (1897). 
t Comptes Rendus, t. cxxvi. p. 997, April 4th, 1898. 
% Astro-Physical Journal, vol. viii. p. 48, June 1898. 

in the Magnetic Field. 169 

B and C may be very close together, or coincide, while the 
centre pair A are separated by a considerable space, as shown 
in fig. 6. In this extreme case we are furnished with a 
triplet in which the centre as it were encloses the sides. 
But this is no specially new form, being quite continuous 
with the other types of modification. 

Fig. 5. 

Fig. 6. 



Once the doubling of the centre line (A, figs. 2 or 4) is 
explained, the other types follow in sequence as expected 
variations, for the cause which converts A into a doublet 
may be sufficiently powerful to separate the constituents of A 
more widely than B is separated from C, and the separation 
of the constituents of A might be tolerably large, even 
though the separation of B and C is quite insensible. 
Thus, if the so-called reversed polarization is shown by any 
lines, the explanation offers no difficulty once we have ex- 
plained the quartet, but it is doubtful if the lines indicated in 
the spectrum of iron by the French and American observers 
just mentioned show this peculiarity. Iron was one of the 
first substances which I examined *, because I considered . it 
might present peculiarities, but I did not observe in it any 
marked differences from the behaviour of other substances. 
Several quartets and other slight modifications occur, but the 
lines referred to by the French and American observers do 
not, on my photographic plates, exhibit the exact peculiarity 
attributed to them. The central part corresponding to A in 
fig. 4 is a doublet without doubt, but the remainder (corre- 
sponding to the lines B and C, fig. 4) does not appear to be by 
any means a single line, but looks rather like a triplet of which 

* See Proc. Koyal Society, January 1898. The doublets referred to in 
this paper turned out on analysis to be quartets. 

170 Dr. T. Prestori on Radiation Phenomena 

the side lines are broad and weak, while the centre is much 
denser. It is just possible, and indeed probable, that these 
modifications may be really quartets, of which the side lines 
B and C are broad and weak, and overlap at their inner 
edges, giving the appearance of a bright central line winged 
with two weaker bands. 

The distance between the side lines B and is about 
the same as that between the components of A, and when the 
double-image prism is not used, the lines in question photo- 
graph as triplets, i. e. as bands having three ribs or denser 
parts running along them lengthwise. With a much stronger 
field it could be determined whether the part BC in these 
lines is really a triplet or an overlapping doublet; but as they 
are all weak lines requiring long exposure (four hours in my 
case), it is not easy to arrange to have a very strong field for 
such a long time. However, it is a matter of very little 
importance at present, for if we can explain the quartet we 
are on the highway to the explanation of all the various 

For this purpose let us revert to Dr. Larmor's paper 
already cited. In this investigation he considers merely the 
simple case of a single ion describing an elliptic orbit under 
a central force directly proportional to the distance. This 
electric charge, when subject to the influence of the magnetic 
field, is so acted on that its elliptic orbit is forced into 
precession round the direction of the magnetic force, that is, 
as a first approximation. For the equations of motion of the 
ion moving round a centre of force in a magnetic field are, 
as a first approximation, the same as those which obtain for 
a particle describing an elliptic orbit under a central force 
when the orbit precesses or revolves round a line through its 
centre drawn in the direction of the lines of magnetic force. 
If N be the natural frequency of revolution of the particle in 
its orbit and n the frequency of revolution of the orbit in its 
precessional movement, the combined movement is equivalent 
to three coexisting motions of frequencies N + w, N, and 
'N — n respectively. When n is small compared with N so 
that its square may be neglected, the equations of motion of 
the particle in the revolving orbit become identical with those 
of the moving ion in the magnetic field. 

This simple theory therefore predicts that a single spectral 
line should be converted into a triplet by the action of the 
magnetic field, and that the constituents of this triplet should 
be plane-polarized when viewed across the lines of force. It 
teaches us that the cause of the tripling is the forced precession 
of the ionic orbits round the lines of magnetic force, and it 

in the Magnetic Field, 171 

assigns a dynamical cause for this precession in the action of 
the magnetic field on the ionic charge moving through it. 

But up to this point the electromagnetic solution deals 
with a perturbation which is really not the full equivalent 
of a precessional movement of the orbit, and therefore the 
investigation given by Dr. Larmor applies, as he himself 
states, to a single simple case. For the equations of motion 
of a particle describing under a central force an elliptic orbit 
which precesses with angular velocity co round a line whose 
direction-cosines are (/, m, n) are 

x=—Q / 2 x + 2(o(ny — mz) + co^x — (o' 2 l(l% + my + nz), . (1) 

with two similar equations for y and z, whereas the equations 
of motion of the ionic charge moving under a central force in 
a magnetic field are, as given by Dr. Larmor, 

x=—£l 2 %-\-k(ny—mz), .... (2) 

with two similar expressions for y and z. The latter equation 
coincides with the former if we neglect co 2 and take 2a>=#, 
that is, when the precessional motion is relatively small. 

The motion imposed on the ion by the electromagnetic 
theory is therefore merely a simple type of precessional pertur- 
bation of the orbit, and, as other perturbations may occur, 
and indeed ought to be expected to occur, it is clear that the 
simple triplet is not the only form which we should expect to 
meet with when the matter is investigated experimentally. 
Thus, if the orbit besides having a precessional motion has in 
addition an apsidal motion, that is a motion of revolution in 
its own plane, then each member of the triplet arising from 
precession will be doubled, and we are presented with a 
sextet as in the case of the D 2 line of sodium. Similarly, if 
the inclination of the plane of the orbit to the line round 
which precession takes place be subject to periodic variations, 
then each member of the precessional triplet will itself become 
a triplet, and so on for other types of perturbation. 

It is quite unnecessary to enter into these matters in any 
detail here, for the whole explanation was fully given and 
published in 1891 by Dr. G. J. Stoney *, that is, six years 
before the effects requiring explanation had been observed. 

Dr. Stoney's aim was to explain the occurrence of doublets 
and equidistant satellites in the spectra of gases, that is in 
the normal spectra unaffected by the magnetic field — for at 

* This most important paper of Dr. Stoney's was published in the 
Scientific Transactions of the Royal Dublin Societ}^ vol. iv. p. 563 
(1891), " On the Cause of Double Lines and of Equidistant Satellites in 
the Spectra of Gases." 

172 Dr. T. Preston on Radiation Phenomena 

that time the influence of the magnetic field was not known 
to exist. The character of certain spectra indicated that the 
lines resolved themselves naturally into groups, or series. For 
example, in the monad elements Na, K, &c. the spectrum 
resolves itself into three series of doublets like the D doublet 
in sodium, and Dr. Stoney's object was to explain the exist- 
ence of these pairs of lines. For this purpose he considered 
what the effect would be on the period of the radiations from 
a moving electron if subject to disturbing forces. In the 
first place he determined that if the disturbing forces cause 
the orbit to revolve in its own plane, that is, cause an apsidal 
motion, then each spectral line will become a doublet. The 
frequencies of the new lines will be N + ft and N — n, where N 
is the frequency of the original line and n the frequency of 
the apsidal revolution. This is very easily deduced by 
Dr. Stoney from the expressions for the coordinates of the 
moving point at any time t. Thus if a particle describes an 
ellipse under a force directed towards its centre (law of direct 
distance), its coordinates at any instant are 

x = a cos fit, y — b sinl2£, 

in which fl is equal to 27rISr, where N is the frequency of 
revolution. But if, in addition, the ellipse revolves around 
its centre in its own plane with an angular velocity co, it is 
easily seen by projection that the coordinates at any time 

x = a cos fit cos cot — b sin fit sin cot, 

y = a cos H t sin cot + b sin fit cos cot , 

and these are equivalent to 

# = i(a + h) cos (fl + co)t + i(a — b) cos (12 — a>)t, 

y — ^{a-[-b) sin (fl + co)t — i(a — 6) sin (fl — co)t, 

and these in turn are equivalent to the two opposite circular 

x x = J(a + b) cos (12 + ©)£ "1 x 2 = i(a — b) cos (12 - »)n 

j/i = i(a + &) sin (fl + co)t J y 2 = — J(a— 6) sin (12 — co)t J ' 

The resultant motion is consequently equivalent to two 
circular motions in opposite senses of frequencies N + n and 
N — n. 

This is an analysis of the motion without any regard to the 
dynamical origin of it ; but if we treat it from a dynamical 
point of view, the equations of motion will exhibit the forces 
which are necessary to bring about the supposed motion. 
Thus, if the orbit rotates with angular velocity o> in its own 

in the Magnetic Field. 173 

plane while the particle is attracted to a fixed centre with a 
force OV 3 then, by taking the moving axes of the orbit as 
axes of reference, the equations of motion are 

x — — £l 2 x + co 2 x + 2coy. \ , „. 

y = — 12^+ coy— 2 cox, J 

so that if (#, y)=e ipt be a solution, we have at once 

p = £l ±co, 

which shows the doubly periodic character of the motion, and 
also exhibits the character of the perturbing forces necessary 
to produce the given apsidal motion of the orbit. For if the 
orbit were fixed, the equations of motion would be (x } y) 
= —£l 2 (x,y): hence the remaining terms on the right-hand 
side of the above equations must represent the perturbing 
forces. Of these the final terms 2 coy and —2 cox are the x 
and y components of a force 2cov, where v is the velocity of 
the particle, acting in a direction perpendicular to v 9 that is 
along the normal to the path of the particle, and represent 
the forces which a charged ion would experience in moving 
through a magnetic field with the lines of force at right 
angles to the plane of the orbit, if 2co be taken equal to k in 
Dr. Larmor's equations (2). The other pair of terms, co 2 x and 
co 2 y, represent a centrifugal force arising from the imposed 
rotation co. If we neglect g> 2 , the above equations become 
identical with those which hold in Larmor's theory for the 
moving ion, as they obviously should, for an apsidal motion in 
the plane of the orbit is the same thing as a precession about a 
line perpendicular to the plane of the orbit, and in this case 
there will be no component in the direction of the axis round 
which precession takes place ; accordingly the middle line of 
the precession triplet will be absent, and we are furnished 
merely with a doublet. 

Now in the magnetic field the perturbing force, being the 
magnetic force, is fixed in direction, and on this account the 
doublets and triplets arising from perturbations caused by it 
are polarized. On the other hand, if the perturbing forces be 
not constant in direction, this polarization should cease to 
exist, and polarization should not be expected in the case of 
any lines of the normal spectrum, even though these happen 
to be derived from other lines by perturbations in the manner 
conceived by Dr. Stoney. 

In the same way the general case of precessional motion 
may be worked backwards in order to discover the types of 
force which produce the perturbation. Thus, taking the axes 

174 Dr. T. Preston on Radiation Phenomena 

moving with angular velocity co round a line whose direction- 
cosines are /, m, n, the equations of motion 

x = — fl 2 x + Zco(ny — mz) + co 2 x — co 2 l(lx-\- my + nz) , 

" = — Q?y + 2co(lz — nx) + co 2 y — co 2 m(lx -J- my + nz), J> (4) 

z = — O 2 ? + 2co(mx — ly) + co 2 z — co' 2 n(lx + my -f nz) 

show that the perturbing forces consist firstly of a force 
2cov sin 6 ; where v is the velocity of the particle and 6 the 
angle which its direction makes with the axis round which the 
precession co takes place. This force acts along the normal to 
the plane of v and co (the direction of the velocity and the axis 
of rotation), and is precisely the force experienced by an 
ion moving in a magnetic field in Larmor's theory. The 
remaining terms containing co 2 are the components of a 
centrifugal force arising from the rotation round the axis 
(I, m, n), and this is negligible only when co is relatively 

If the direction I, m, n be taken to be that of the lines of 
magnetic force, and if the axis of z be taken to coincide with 
this direction, then the equations (4) simplify into 

x=-(Q?-co' ;L )x + 2coy~\ 
y=-(W-co*)ij-2cox\ (5) 

s = -n% j 

and these are the equations of motion of a particle describing 
an elliptic orbit which precesses with angular velocity co round 
the axis of z. The two first of these equations contain x and 
y and give the projection of the orbit on the plane x, y, at 
right angles to the axis of the magnetic field. This pro- 
jection is an ellipse revolving in its own plane with an apsidal 
angular velocity co, and gives rise to the two side lines of the 
normal triplet of frequencies (£1 ± co)I'Itt. On the other hand, 
the vibration parallel to the axis of z is unaffected by the 
precessional motion and gives rise to the central line of the 
triplet of frequency H/2ir. 

Now in order to account for the quartet (fig. 2) we must 
introduce some action which will double the central line A 
while the side lines B and C are left undisturbed. That is, 
we must introduce a double period into the last of equations 
(5) while the first and second remain unchanged. This is 
easily done if we write the equation for z in the form 

2 = AsinX2£ . . . . . ... (6) 

in the Magnetic Field. 175 

and remark that this will represent two superposed vibrations 
of different periods, if we regard A as a periodic function of 
the time instead of a constant. That is, if we take A to be 
of the form a sin nt, we shall have 

z = a sin nt sin Clt = -p[GOS (ft — n)t — cos {Q, + n)t] } 

which represents two vibrations of equal amplitude and of 
frequencies (ft — ?i)/2tt and (ft + w)/27r as required to produce 
the quartet. The magnitude of n determines whether the 
separation of the constituents of the central line A (fig. 2) 
shall be less than, or greater than, the separation of the side 
lines B and C, and if the former is sensible while the latter is 
insensible we are presented with the case depicted in fig. 6 — 
although, as I have said before, my observations do not 
confirm the existence of this case. 

The supposition made above to account for the doubling of 
the middle line, viz. that the amplitude of the z component of 
the vibration varies periodically, is one which appears to be 
justified when we consider the nature of the moving system and 
the forces which control it. For the revolving ion is part of 
some more or less complex system which must set in some 
definite way under the action of the magnetic field — say with 
its axis along the direction of the magnetic force — and, in 
coming into this position, the inertia of the system will cause it 
to vibrate with small oscillations about that position of equi- 
librium, and this vibration superposed on the precessional 
motion of the ionic orbit gives the motion postulated above 
to explain the quartet. 

This, indeed, comes to the same thing as a suggestion made 
by Professor Gr. F. FitzGrerald about a year ago — shortly 
after I discovered the existence of the quartet form (Oct. 1897). 
In Professor Fitz Gerald's view, the ion revolving in its orbit is 
equivalent to an electric current round the orbit, and therefore 
the revolving ion and the matter with which it is associated 
behave as a little magnet having its axis perpendicular to the 
plane of the orbit. The action of the magnetic field will be to 
set the axis of this magnet along the lines of force, and in 
taking up this set the ionic orbit will vibrate about its position 
of equilibrium just as an ordinary magnet vibrates about its 
position of rest under the earth's magnetic force. 

In a similar way a periodic change in the eliipticity of the 
orbit produces a doubling of the lines, while a periodic 
oscillation in the apsidal motion renders the line nebulous or 
diffuse; and by treating these cases in the foregoing manner 
■the corresponding forces may be discovered. It is clear, 

176 Dr. T. Preston on Radiation Phenomena 

therefore, that perturbations of this kind are sufficient to 
account for all the observed phenomena, and, further, that 
perturbations of this kind are almost certain to be in 
operation throughout some, at least, of the ionic motions. 

The existence of all these variations of the normal triplet 
type are therefore of great interest, not only in showing that 
the perfect uniformity required for the production of the 
normal triplet is not maintained, as we should expect, in all 
cases, but also as an experimental demonstration that the 
causes supposed by Dr. Stoney, in 1891, to be operative in 
producing doublets and satellites in the natural spectra of 
gases may be really the true causes by which they are produced. 

Nevertheless Dr. Stoney's explanation of the natural doublets 
is opposed by a serious difficulty in the fact that the two lines 
of a given doublet, say the two D lines of sodium, behave in 
different ways, as if they arose from different sources rather 
than from the perturbation of the same source. For, in 
addition to the differences previously known to exist, there is 
the difference of behaviour in the magnetic field. Thus T) x 
is a wide-middled quartet in which the distance between the 
central lines A (fig. 4) is nearly as great as the distance 
between the side lines B and C, while D 2 shows as a sextet of 
uniformly spaced lines. 

In a similar manner individual members of the natural 
triplets which occur in the natural spectra of the zinc, 
cadmium, magnesium, &c. group behave differently. Thus 
if we denote the members of one of the natural triplets 
by the symbols T 1? T 2 , T 3 , in ascending order of re Iran - 
gibility (for example the triplet 5086, 4800, 4678 of 
cadmium, or the triplet 4811, 4722, 4680 of zinc, or the 
green b triplet of magnesium), we find that T 3 in all cases, in 
the magnetic field, shows as a pure triplet, or suffers accord- 
ing to the foregoing merely precessional perturbation. On 
the other hand, T 2 shows in each case as a quartet while T 1 is 
a diffuse triplet in which each of the members may prove to 
be complex on further resolution. This would seem to point 
to an essential difference in the characters of the lines T 1? T 2 , T 3 , 
as if they sprang from different origins rather than immedi- 
ately from the same. It is also of great interest to note that, 
so far as my observations yet show, these natural triplets 
behave differently according as they belong to Kayser and 
Runge's first subsidiary series or to the second subsidiary 
series. Thus if the triplet T 1? T 2 , T 3 , belongs to the first 
subsidiary series, then the magnetic effect decreases from T 2 
to T 3 , while if it belongs to the second subsidiary series, the 
magnetic effect increases from T x to T 3 . Examples of this 

in the Magnetic Field. 177 

latter class are shown in my communication to this Journal 
(April 1898, p. 335), where the increasing character of the 
magnetic effect is well exhibited in the natural triplets 5086, 
4800, 4678 of cadmium, and 4811, 4722, 4680 of zinc. 
Further examples of this, and other peculiarities, I hope to 
give in the near future as soon as I have fully examined and 
verified them. 

General Law. 

The first general survey of the magnetic effect on the 
spectral lines of any given substance did not appear to favour 
the view that the phenomena are subject to any simple law. 
According to the electromagnetic theory the separation, S\, 
of the side lines of a magnetic triplet should, under the same 
conditions, vary directly as X 2 as we pass from line to line of 
the same spectrum. The possibility of such a law as this 
seemed to be refuted by the fact that some lines are largely 
affected in the magnetic field while others, of nearly the same 
wave-length in the same spectrum, are not appreciably affected 
under the same circumstances. In this connexion, however, 
I pointed out * that " it is possible that the lines of any one 
substance may be thrown into groups for each of which hX 
varies as A. 2 , and each of these groups might be produced by 
the motion of a single ion. The number of such groups in a 
given spectrum would then determine the number of different 
kinds of ions in the atom or molecule. 

" Homologous relations may also exist between the groups 
of different spectra, but all this remains for complete in- 

Although the investigation referred to in the foregoing is 
still far from complete, yet the measurements so far made 
uniformly tend to confirm the above speculation. For the 
corresponding lines of the natural groups into which a given 
spectrum resolves itself possess the same value of e/m or 8X/X 2 3 
and, further, this value is the same for corresponding lines in 
homologous spectra of different substances. 

To illustrate the meaning of this, take the case of mag- 
nesium, cadmium, and zinc, which are substances possessing 
homologous spectra and belonging to the same chemical 
group (MendelejefFs second group). The spectra of these 
metals consist of series of natural triplets. The first triplet 
of the series in magnesium is the green b group consisting 
of the wave-lengths 5183*8, 5172-8, 5167'5 ; while the first 
cadmium triplet consists of the lines 5086, 4800, 4678, and 

Phil. Mag., April 1898, p. 337. 

1 78 On Radiation Phenomena in the Magnetic Field. 

the first zinc triplet consists of the lines 4810*7, 4722, 4680. 
Each of these triplets belongs to Kayser and RungVs second 
Nebenserie, being the first terms, corresponding to n = 3, in 
their formula. We should consequently expect these groups 
to behave similarly in the magnetic field and to show effects 
which are similar for corresponding lines. That this expect- 
ation is realized is shown by the following table : — 



Zinc. -mje or \ 2 /8\. 




4810-7 18 approx. 

Diffuse triplets. 



4722 11-5 „ 




4680 10 

Pure triplets. 

Thus the corresponding lines 5183'8, 5086, and 4810*7 of 
the different substances possess the same value for m/e, while 
the other corresponding lines also possess a common value 
for the quantity m/e. The value of this quantity changes 
from one set of lines to another, showing, as we should expect, 
that the different sets arise from differences in the source 
which produces them. 

Not only is the quantity m/e the same for corresponding 
lines in homologous spectra, but, as shown in the above table, 
the character of the magnetic effect is also the same for 
corresponding lines. Thus, while the lines along the lowest 
row, 5167-5, 4678, 4680, are all of the pure triplet type, the 
lines of the middle row all become resolved into similar 
quartets in the magnetic field, and the lines forming the top 
row are all somewhat diffuse and show as " soft " triplets of 
which the constituents may be really complex on further 

It thus appears that the observation of radiation pheno- 
mena in the magnetic field is likely to afford a valuable means 
of inquiry into the so far hidden nature of the events which 
bring about the radiation from a luminous body, and also to 
give us, perhaps, some clearer insight into the structure of 
matter itself. 

[ 179 ] 

XII. On the Reflexion and Refraction of Solitary Plane 
Waves at a Plane Interface behveen two Isotropic Elastic 
Mediums — Fluid, Solid, or Ether . By Lord Kelvin, 

§ 1. "rC'LASTIC SOLID" includes fluid and ether; except 
J__i conceivable dynamics f of the mutual action 
across the interface of the two mediums. Maxwell's electro- 
magnetic equations for a homogeneous non-conductor of 
electricity are identical with the equations of motion of an 
incompressible elastic solid J, or with the equations expressing 
the rotational components of the motion of an elastic solid 
compressible or incompressible ; but not so their application 
to a heterogeneous non-conductor or to the interface between 
two homogeneous non-conductors §. 

§ 2. The equations of equilibrium of a homogeneous elastic 
solid, under the influence of forces X, Y, Z, per unit volume, 
acting at any point (x, y, z) of the substance are given in 
Stokes' classical paper u On the Theories of the Internal 
Friction of Fluids in Motion, and of the Equilibrium and 
Motion of Elastic Solids/' p. 115, vol. i. of his ' Mathematical 
Papers': also in Thomson and Tait's ' Natural Philosophy' 
[§698(5) (6)]. Substituting according to D'Alembert's 
principle, — pf, — py, —pi for X, Y, Z, and using as in 
a paper of mine |] of date Nov. 28, 1846, V 2 to denote 

d 2 d l d 2 
the Laplacian operator -7-3 + j— 2 + 'J^-ii we ^ n< ^ as the equa- 
tions of motion x & z 

pS =( - k+ ° n)d £ +nv ^ 

* Communicated by the Author j having- been read before the Royal 
Society of Edinburgh on December 19, 1898. 

t See Math, and Phys. Papers, vol. iii. art. xcrx. (first published 
May 1890), §§ 14-20, 21-28; and particularly §§ 44-47. Also Art. c. 
of same volume ; from Comptes Rendus for Sept. 16, 1889, and Proc 
Roy. Soc. Edinb., March 1890. 

\ See ' Electricity and Magnetism,' last four lines of § 616, last four 
lines of § 783, and equations (9) of § 784. 

§ Ibid. §611, equations (1*). In these put 0=0, and take in con- 
nexion with them equations (2) and (4) of § 616. Consider K and u as 
different functions of x, y, z ; consider particularly uniform values for 
each of these quantities on one side of an interface, 'and different uniform 
values on the other side of an interface between two different non- 
conductors, each homogeneous. 

|| Camb. and Dublin Math. Journal, vol. ii. (1847). Republished as 
Art. xxvn., vol. i. of Math, and Phys. Papers. 

180 Lord Kelvin on the Reflexion 

p denoting the density of the medium, f , rj, f its displacement 
from the position of equilibrium (#,y,s), and 8 the dilatation 
of bulk at (a, y, z) as expressed by the equation 

*-2?J+2 0* 

§ 3. Taking d/dx, d/dy, d/dz of (1), we find 

d' 2 S 
pjr== (*+*»)?* • (3). 

From this we find 

v -* 8 = Ltfc.(Jp (4 , 

Put now 

e-«H-£v*i • -* + £ V- 2 S; l-6 + £^I (5). 
These give 

fi + p + f = (6) 

and therefore, eliminating by them £. 9;, £ from (1), we find 
by aid of (4), 

p d S= n ^> p c w= n ^> p d S= n ^-> w- 

§ 4. By Poisson's theorem in the elementary mathematics 
of force varying inversely as the square of the distance, 
we have 

V~ 2 S = -|^; J J J d (volume) • pp ; (8) , 

where S, 8' denote the dilatations at auy two points P and P ; ; 
d (volume) denotes an infinitesimal element of volume around 
the point P x ; and PP' denotes the distance between the points 
P and P'. This theorem gives explicitly and determinately 
the value of V _2 8 for every point of space when o is known 
(or has any arbitrarily given value) for every point of space. 
§ 5. If now we put 

f2 =j-v- 2 s; % =|v- 2 a ; &=|v-a ; (9) , 

we see by (5) that the complete solution of (1) is the sum of 
two solutions, (£ 1? %, fi) satisfying (6) and therefore purely 
distortional without condensation; and (f 2 , %, f 2 ), ivhich, in 

and Refraction of Solitary Plane Waves. 181 

virtue of (9) , is irrotational and involves essentially rarefaction 
or condensation or both. This most important and interesting 
theorem is, I believe, originally due to Stokes. It certainly 
was given for the first time explicitly and clearly in § § 5-8 
of his " Dynamical Theory of Diffraction '"'*. 

§ 6. The complete solution of (3) for plane waves travelling 
in either or both directions with fronts specified by (a, ft 7), 
the direction-cosines of the normal, is, with ty and % to denote 
arbitrary functions, 

s = ^(,_^±to) +x (, + f£±to) (10) , 


e=?v /*+t? (11) . 

so that v denotes the propagational-velocity of the con- 
den sational-rarefactional waves. By inspection without the 
aid of (8), we see that for this solution 

v^-K!H*H**^ + x(. + =**±*!)i] 

For our present purpose we shall consider only waves 
travelling in one direction, and therefore take %=0; and, 

(d \ — * 
T.J f 

instead of v( -rj ^ ;f being an arbitrary function. Thus 

by (12) and (9) we have, for our condensational-rarefactional 

k_ 7 h__k_, f( t gg±gy±3? \ . . (13). 
* ft y J \ V J 

In the wave-system thus expressed the motion of each 
particle of the medium is perpendicular to the wave- front 
(a, ft 7). For purely distortional motion, and wave-front 
still (a, ft 7) and therefore motion of the medium everywhere 
perpendicular to (a, ft 7), or in the wave-front, we find 
similarly from (7) and (6) 

fi - vi _ Sl rls ax+ fy+v z \ (u), 

^A" i 8B-7C" / V u / ' 


••■V7 (15) ' 

* Camb. Phil. Trans., Nov. 26, 1849. Republished in vol. ii. of his 
' Mathematical Papers.' 

Phil. Mag. S. 5. Vol. 47. No. 285. Feb. 1899. 

182 Lord Kelvin on the Reflexion 

and so denotes the propagational velocity of the distortional 
waves ; and A, B, C are arbitrary constants subject to the 

«A + /3B + 7C = (16). 

§ 7. To suit the case of solitary waves we shall suppose the 
arbitrary function f(t) to have any arbitrarily given value for 
all values of t from to t, and to be zero for all negative 
values of t and all positive values greater than t. Thus t is 
what we may call the transit-time of the wave, that is, the 
time it takes to pass any fixed plane parallel to its front ; or 
the time during which any point of the medium is moved by 
it. The thicknesses, or, as we shall sometimes say, the wave* 
lengths, of the two kinds of waves are ur and vr respectively, 
being for the same transit-times directly as the propagational 

§ 8. And now for cur problem of reflexion and refraction. 
At present we need not occupy ourselves with the case of 
purely distortional waves with vibratory motions perpendicular 
to the plane of the incident, reflected, and refracted rays. It 
was fully solved by Green * with an arbitrary function to 
express the character of the motion (including therefore the 
case of a solitary wave or of an infinite procession of simple 
harmonic waves). He showed that it gave precisely the 
" sine law " which Fresnel had found for the reflexion and 
refraction of waves " polarized in the plane of incidence." 
The same law has been found for light, regarded as electro- 
magnetic waves of one of the two orthogonal polarizations, 
by von Helmholtz, H. A. Lorenz, J. J. Thomson, Fitz Gerald, 
and Rayleigh f . None of them has quite dared to say that 
the physical action represented by his formulas for this case is 
a to-and-fro motion of the ether perpendicular to the plane of 
incidence, reflexion, and refraction ; nor has any one, so far 
as I know, absolutely determined whether it is the lines of 
electric force or of magnetic force that are perpendicular 
to that plane in the case of light polarized by reflexion at the 
surface of a transparent medium. For the action, whatever 
its rjhysical character may be, which takes place perpendicular 
to that plane, they all seem to prefer " electric displacement/' 
of which the only conceivable meaning is motion of electricity 
to and fro perpendicular to the plane. If they had declared, 
or even suggested, definitely this motion of ether, they would 

* " On the Reflexion and Refraction of Light at the common Surface 
of two Non-Crystallized Media/' Math. Papers, p. 258. Also Trans. 
Caruo. Phil. Soc. 1838. 

t Sde Glazebrook's Rsport " on Optical Theories " to British Asso- 
ciation, 1885. 

and Refraction of Solitary Plane Waves. 


have been perfectly in harmony with the undulatory theory of 
light as we have it from Young and Fresnel. We shall return 
to this very simple problem of reflexion and refraction of 
purely distortional waves in which the motion is perpendicular 
to the plane of the three rays, in order to interpret in the very 
simplest case the meaning, for a solitary wave, of the " change 
of phase " discovered by Fresnel and investigated dynamically 
by Green for a procession of periodic waves of simple harmonic 
motion experiencing " total internal reflexion/' (See § 20 

§ 9. Meantime we take up the problem of the four reflected 
and refracted waves produced by a single incident wave of 
purely distortional character, in which the motion is in a 
plane perpendicular to the five wave-fronts. Taking this for 
XOY, the plane of our diagram, let YOZ be the interface 
between the two mediums. We shall first consider one 
single incident wave, I, of the purely distortional character. 
By incidence on the interface it will generally introduce 
reflected and refracted waves 1', I y , of its own kind, that is 
purely distortional, and J', J r reflected and refracted waves 

Fm. 1, 

of the condensational-rarefactional kind. The diagrams re- 
present, for two cases, sections of portions of the five waves 
by the plane XOY. F and R show the front and rear of 
each wave ; and the lines of shading belonging to it show 
the direction of the motion, or of the component, which it 



Lord Kelvin on the Reflexion 

gives to the medium. The inclinations of the fronts and rears 
to OX, being what are ordinarily called the angle of incidence 

FiK. 2. 

and the angles of reflexion or refraction of the several waves, 
will be denoted by i, i', i n j,j r The value of 7 for each of 
the five waves is zero, and the values of a and ft are as shown 
in the following table :— 




-j- sin i 

— cos i 


+ sin i 

+ cos i 
— cos i, 


+ sin i t 


+ siiy" 

+ cos^' 


+ sinj?; 

- cosj>*, 

The section of the five waves by OX is the same for all, 
being expressed by ur/sin i for I, and by corresponding 
formulas for the four others. Hence if we denote r-times its 

and Refraction of Solitary Plane Waves. 185 

reciprocal by a, we have 

a _ sin i _ sin /, __ shy' _ sin,/, 

where u and ?/, are the propagational velocities of the distor- 
tional waves, and v, v n those of the condensational waves in 
the two mediums. If now we take 

b = acoti= \Z(ii- 2 —a 2 ) ; b l = acoti l = s/(u~ 2 — a 2 ) ; 
c=acotj= \Z(v~ 2 — a 2 ) ; e, = acotj t = \/(v t ~ 2 — « 2 ); (18), 

we have for the arguments of/ in the five waves 

t— ax + by; t—ax — by; t — ax + b t y; t — ax — cy; t — ax + Cjy (19), 

§ 10. Following Green * in calling the two sides of the 
interface the upper and lower medium respectively (and so 
shown in the diagram), we have for the components of the 
displacement in the upper medium 

g=blf(t — ax + hy)—blf (t — ax — by) + aJ'f (t — ax — cy) | 

r) = alf(t — ax + by)+al'f(t — ax-by) +cJ'f[t — ax-cy) J ^' '' 

and in the lower medium 

^bLfit-ax+b^-raJJit-ax + cy) | 
rj = al,f(t — ax + b,y) — c t JJ(t — ax+ c,y) ) 

where I, I', I,, J', J p denote five constant coefficients. The 
notation J' and J, is adopted for convenience, to reserve the 
coefficient J for the case in which the incident wave is con- 
densational, and there is no incident distortional wave. 
There would be no interest in treating simultaneously the 
results of two incident waves, one distortional (I) and the 
other condensational (J). 

§ 11. We may make various suppositions as to the inter- 
facial conditions, in respect to displacements of the two 
mediums and in respect to mutual forces between them. 
Thus we might suppose free slipping between the two : that 
is to say, zero tangential force on each medium; and along 
with this we might suppose equal normal components of 
motion and of force ; and whatever supposition we make as 
to displacements, we may suppose the normal and tangential 
forces on either at the interface to be those calculated from 
the strains according to the ordinary elastic solid theory, or 
to be those calculated from the rotations and condensations or 
dilatations, according to the ideal dynamics of ether suggested 
in the article referred to in the first footnote to § 1. We shall 

* Green's ' Math. Papers/ p. 253. 


186 Lord Kelvin on the Reflexion 

for trie present take the case of no interfacial slip, that is, equal 
values of g, 7] on the two sides of the interface. Remarking 
now that the argument of /for every one of the five waves is 
t — ax where y = 0, we see that the condition of equality of 
displacement on the two sides of the interface gives the 
following equations : — ■ 

6(1-1') +oJ / =& / I / + aJ i TJ 

a (I + 1') + c J' = al J — c l J l f 

§ 12. As to the force-conditions at the interface, I have 
already given, for ordinary elastic solid or fluid matter * on 
the two sides of the interface, a complete solution of the 
present problem in my paper f "On the Eeflexion and 
Refraction of Light " in the ' Philosophical Magazine ' for 
1888 (vol. xxvi.) ; nominally for the case of simple harmonic 
wave-motion, but virtually including solitary waves as 
expressed by an arbitrary function: and I need not now 
repeat the work. At present let us suppose the surface-force 
on each solid to be that which I have found it must be 
for ether J, if magnetic force is due to rotational displace- 
ment of ether, and the lines of magnetic force coincide with 
axes of rotation of etherial 'substance. According to this 
supposition the two components, Q (normal) and T (tan- 
gential), of the mutual force between the mediums, which 
must be equal on the two sides of the interface, are 

Q=«(f + ?) 

\dx dy) ™ 

\dx dy) '" J \dx dy. 

where k denotes for ether that which for the elastic solid we 
have denoted by (k +■ fft), and suffixes indicate values for the 
lower medium. If we begin afresh for ether, we may define 
n as l/4«r of the torque required to hold unit of volume 
of ether rotated through an infinitesimal angle us from its 

* The force-conditions for this case are as follows : — 

Normal cooiponent force equated for upper and lower mediums, 

C*-f.)«+a.5=(t -•»>,+*., (J) /S 

and taDg-ential forces equated, 


t In that paper B, A, and £ denote respectively the n, the k-\-pi, and 
the p of the present paper. 
J See first footnote to § 1. 

and Refraction of Solitary Plane Waves. 187 

orientation of equilibrium, and k as the bulk-modulus, that is 
to say, the reciprocal of the compressibility, of ether. Thus 
we now have as before in equations (15), (11), and (18) 

P • ^2. J^ — 1,-2— P 

a 2 + b 2 = u~ 2 =^; a 2 + b 2 = u { 

a 2+ c 2 = v -2 = P_. a 2 + c 2 = v -2-P 


K Kj 

Using (20) and (21) in (23) with y = we nod 

n(a? + b 2 )(I + I') =71^ + 0?)!, J ' * [ ] ' 
whence by (24) 

pJ'=p,J<; p(i+i')=M; (26). 

By these equations eliminating \ and J, from (22), we find 
-{bp J -b J p)l+(bp l + b l p)V=a{p l --p)Z l 

X,,} ■ ■ ■ m- 

«0»-p)(I+I') = -(cp 

and solving these equations for 1' and J' in terms of I, w 

jr_ (bp-b^jcpt + cp) -a 2 (p-p) 2 j 
{bp l + bfi) {cp, + c t p) + a- (p-p) 2 , 

j, = -2ab Pl {p-p) • * 

^P,+ b iP) {cpi + W) +a 2 (Pi-py 

and with J' and V thus determined, (26) give J y and I,, 
completing the solution of our problem. 

§ 13. Using (18) to eliminate a, b, b n c, and c t , from (28), 
and putting 

, P/ ~ P ,. =h (29); 

we find 

r _ p, cot i— p cot i-h(p-p) 

I /^coU + pcot^ + A^-p) ^ oU ^ 


J /_ -2/^ cot? 

. . . (ol). 

I p, voti + p cot ij-\- h(p j— p) 

the case of t? and ^ y ve 
ad Uj\ which by (2S) make 

cotj^l/va, and coty y = l/v y a . . . (32). 

Consider now the case of v and v, very small in compa- 
rison with u and w y ; which by (28) makes 

188 Lord Kelvin on the Reflexion 

This gives 

h ^ (p-p) S ini (33) 

r/ v r V, 

which is a very small numeric. Hence J 7 is very small in 
comparison with I ; and 

I' ^ p, cot i-p cot ij ^ (M . 

1 * pi cot i + p cot i t 

§14. If the rigidities of the two mediums are equal, we 
have p / \p = sin 2 i \ sin 2 i n and (34) becomes 

V _ sin 2i— sin 2i / _ tan (i—ij) ,„~. 

I ~~ sin 2i + sin 2i y ~ tan (i + i y ) ^ v' 

which is Fresnel's " tangent-formula." On the other hand, 
if the densities are equal, (34) becomes 

I'_ —gin ft'— t,) (9C] 

I~ sin(t + t,) •.•••• WW. 

which is FresnePs " sine-formula " ; a very surprising and 
interesting result. It has long been known that for vibrations 
perpendicular to the plane of the incident, reflected, and 
refracted rays, unequal densities with equal rigidities of the 
two mediums, whether compressible or incompressible, gives 
Fresnel's sine-law : and unequal rigidities, with equal 
densities, gives his tangent-law. But for vibrations in the 
plane of the three rays, and both mediums incompressible, 
unequal rigidities with equal densities give, as was shown by 
Rayleigh in 1871*, a complicated formula for the reflected 
ray, vanishing for two different angles of incidence, if the 
motive forces in the waves are according to the law of 
the elasticity of an ordinary solid. Now we find for vibrations 
in the plane of the rays, Fresnel's sine-law, with its continual 
increase of reflected ray with increasing angles of incidence 
up to 90°, if the restitutional forces follow the law of 
dependence on rotation which I have suggested f for ether, 
and if the waves of condensation and rarefaction travel at 
velocities small in comparison with those of waves of dis- 

§15. Interesting, however, as this may be in respect to an 
ideal problem of dynamics, it seems quite unimportant in the 
wave-theory of light ; because Stokes J has given, as I 

* Phil. Mag. 1871, 2nd half year. 

t " On the Reflexion and Refraction of light," Phil. Mag. vol. xxvi. 


% "' Dynamical Theory of Diffraction." See footnote §5. 

and Refraction of Solitary Plane Waves. 189 

believe, irrefragable proof that in light polarized by reflexion 
the vibrations are perpendicular to the plane of the incident 
and reflected rays, and therefore, that it is for vibrations in 
this plane that Fresnel's tangent-law is fulfilled. 

§ 16. Of our present results, it is (35) of §14 which is really 
important ; inasmuch as it shows that Fresnel's tangent-law 
is fulfilled for vibrations in the plane of the rays, with the 
rotational law of force, as I had found it in 1888 * with the 
elastic-solid-law of force, provided only that the propagational 
velocities of condensational w T aves are small in comparison 
with those of the waves of transverse vibration which 
constitute light. 

§17. By (28) we see that when a~ l , the velocity of the 
wave-trace on the interface of the two mediums, is greater 
than the greatest of the wave-velocities, each of b, b n c, Cj is 
essentially real. A case of this character is represented by 
fig. 2, in which the velocities of the condensational waves 
in both mediums are much smaller than the velocity of the 
refracted distortion al wave, and this is less than that of the 
incident wave which is distortional. When one or more of 
b, b /y c, Cy is imaginary, our solution (26) (28) remains valid, 
but is not applicable to /regarded as an arbitrary function ; 
because although f(t) may be arbitrarily given for every real 
value of t, we cannot from that determine the real values of 

f(t +l g)+f(t- lq ) (37), 

»{/(* + «?)-/(«-«?)} .... (38). 

The primary object of the present communication was to 
treat this case in a manner suitable for a single incident soli- 
tary wave whether condensational or distortional ; instead of 
in the manner initiated by Green and adopted by all subsequent 
writers, in which the realized results are immediately applicable 
only to cases in which the incident wave-motion consists of 
an endless train of simple harmonic waves. Instead, therefore, 
of making / an exponential function as Green made it, I 

to-thz < 39 )' 

where r denotes an interval of time, small or large, taking the 
place of the " transit-time " (§7 above), which we had for the 
case of a solitary wave-motion starting from rest, and coming 
to rest again for any one point of the medium after an interval 
of time which we denoted by t. 

* See footnote §14. 

190 On Reflexion and Refraction of Solitary Plane Waves. 
§18. Putting now 

I=p + iq (40); 

and from this finding V, I n J', J / ; and taking for the real 
incident wave-motion (§10 above) 

f = V _ i r p + iq p-ig -| "| 

b a 2 Lt—ax + by + iT t — ax + by — trj \ 

_ p(t — ax + by) + qT | 

(t-ax + by)* + T 2 J 

being the mean of the formulas for -f 1 and —t; we find a 
real solution for any case of b n c, c n some or all of them 

§19. Two kinds of incident solitary wave are expressed by 
(41), of types represented respectively by the following 
elementary algebraic formulas : — 

t-ax + by 


{t—ax + byf + i* 

(t-ax + byf + T* 


The same formulas represent real types of condensational 
waves with f/a and y/( — c), instead of the f/6 and rj/a of (41) 
which relates to distortional waves. It is interesting to 
examine each of these types and illustrate it by graphical 
construction : and particularly to enquire into the distribution 
of energy, kinetic and potential, for different times and places 
in a wave. Without going into details we see immediately 
that both kinetic and potential energy are very small for any 
value of (t—ax + by) 2 which is large in comparison with t 2 . 
I intend to return to the subject in a communication 
regarding the diffraction of solitary waves, which I hope to 
make at a future meeting. 

§20. It is also very interesting to examine the type- 
formulas for disturbance in either medium derived from (41) 
for reflected or refracted waves when b n or c, or c/ is 
imaginary. They are as follows, for example if b ; = ty, where 
g is real ; 

t ^ ax ... . (44) 

(t-ax) 2 + (gy + T) 2 { h 

9y + T ... (45). 

(t-ax)*+(gy + r) 

Prof. H. L. Calleudar on Platinum Thermometry. 191 

These real resultants of imaginary waves are not plane 
waves. They are forced linear waves sweeping the interface, 
on which they travel with velocity a~ ] ; and they produce 
disturbances penetrating to but small distances into the 
medium to which they belong. Their interpretation in con- 
nexion with total internal reflexion, both for vibrations in the 
plane of the rays, and for the simpler case of vibrations 
perpendicular to this plane (for which there is essentially no 
condensational wave) constitutes the dynamical theory of 
FresnePs rhomb for solitary waves. 

XIII. Notes on Platinum Thermometry. By H. L. Cal- 
LEKDAK, M.A., P.P.S., Quain Professor of Physics, Uni- 
versity College , London*. 

SINCE the date of the last communication, which I made to 
this Journal in February 1892, I have been continually 
engaged in the employment of platinum thermometers in 
various researches. But although I have exhibited some of 
my instruments at the Royal Society and elsewhere and have 
described the results of some of these investigations, I have 
not hitherto found time to publish in a connected form an 
account of the construction and application of the instruments 
themselves, or the results of my experience with regard to the 
general question of platinum thermometry. As the method 
has now come into very general use for scientific purposes, it 
may be of advantage at the present time to collect in an 
accessible form some account of the progress of the work, to 
describe the more recent improvements in methods and 
apparatus, and to discuss the application and limitations of 
the various formulas which have from time to time been 

The present paper begins with a brief historical summary, 
with the object of removing certain common misapprehensions 
and of rendering the subsequent discussion intelligible. It 
then proceeds to discuss various formulas and methods of 
reduction, employing in this connexion a proposed standard 
notation and nomenclature, which I have found convenient in 
my own work. I hope in a subsequent paper to describe some 
of the more recent developments and applications of the 
platinum thermometer, more particularly those which have 
occurred to me in the course of my own work, and which have 
not as yet been published or described elsewhere. 

* Communicated by the Author. 

192 Prof. H. L. Callendar on Platinum Thermometry, 
Historical Summary. 

The earlier experiments on the variation of the electrical 
resistance of metals with temperature were either too rough, 
or too limited in range, to afford any satisfactory basis for a 
formula. The conclusion of Lenz (1838), that the resistance 
reached a maximum at a comparatively low temperature, 
generally between 200° and 300° C, was derived from the 
empirical formula, 

R°/R=l + at'+bt 2 , (L) 

in which R° and R stand for the resistances at 0° and t° C, 
respectively. This conclusion resulted simply from the 
accident that he expressed his results in terms of conductivity 
instead of resistance, and could be disproved by the roughest 
qualitative experiments at temperatures beyond the range 
0° to 100° C, to which his observations were restricted. 
Matthiessen (1862), in his laborious and extensive investiga- 
tions, also unfortunately fell into the same method of 
expression. His results have been very widely quoted and 
adopted, but, owing to the extreme inadequacy of the formula, 
the accuracy of his work is very seriously impaired even 
within the limits of the experimental range to which it was 
confined. The so-called Law of Glausius, that the resistance 
of pure metals varied as the absolute temperature, was a 
generalization founded on similarly incomplete data. The 
experiments of Arndtsen (1858), by which it was suggested, 
gave, for instance, the temperature-coefhcients '00394 for 
copper, '00341 for silver and *00413 for iron, all of which 
differ considerably from the required coefficient '003G65. 
The observations, moreover, were not sufficiently exact to 
show the deviation of the resistance-variation from lineality. 
The experiments of Sir William Siemens (1870) did not 
afford any evidence for the particular formula which he pro- 
posed, at least in the case of iron. These formulas have been 
already discussed in previous communications'^, but con- 
sidering the extent to which they are still quoted, it may be 
instructive to append the curves representing them, as a 
graphic illustration of the danger of applying for purposes of 
extrapolation formulas of an unsuitable type. The curves 
labelled Morrisf and Benoit, which are of the same general 
character but differ in steepness, may be taken as representing 
approximately the resistance-variation of specimens of pure 
and impure iron respectively. 

The first experiments which can be said to have afforded 
any satisfactory basis for a general formula were those of 

* Callendar, Phil. Mag. July 1891 ; G. M. Clark, Electrician, Jan. 1897. 
t Phil. Mag. Sept. 1897, p. 213. 

Prof. H. L. Callendar on Platinum Thermometry. 193 

Benoit (Comptes JRendus, 1873, p. 342). Though apparently 
little known and seldom quoted, his results represent a great 
advance on previous work in point of range and accuracy. 

Fio-. 1. 



Em pi 




ae - 


















{ f 

































—300° — 2JG° -100 100° 200° 300° 400 J 500° 600° 700° 300° 900° 000° 

The wires on which he experimented were wound on clay 
cylinders and heated in vapour-baths of steam (100°), mercury 
(360°), sulphur (440°), and cadmium (860°), and in a liquid 
bath of mercury for temperatures below 360°. The resistances 
were measured by means of a Becquerel differential galvano- 
meter and a rheostat consisting of two platinum wires with a 
sliding mercury-contact. It is evident that the values which 
he assumed for the higher boiling-points are somewhat rough. 
The boiling-point attributed to cadmium, following Deville 
and Troost, is about 50° too high according to later experi- 
ments by the same authorities, or about 90° too high according 
to Carnelley and Williams. It would appear also that no 
special precautions were taken to eliminate errors due to 
thermoelectric effects, to changes in the resistance of the 
leading wires, and to defective insulation, &c. In spite of 
these obvious defects it is surprising to find how closely the 
results as a whole agree with the observations of subsequent 
investigators. The resistance-variation of all the more common 

194 Prof. H. L. "Callendar on Platinum Thermometry. 

metals, according to Benoit, is approximately represented by 
an empirical formula of the type 

R/R°=l + at + bt 2 , (B) 

where B is the resistance at any temperature t, and R° the 
resistance at 0° C. The values of the constants a and b which 
he gives for iron and steel represent correctly (in opposition 
to the formula of Siemens) the very rapid increase in the rate 
of change of resistance with temperature, as shown by the 
relatively large positive value of the coefficient b. He gives 
also in the case of platinum a small negative value for b (a 
result since abundantly confirmed), although the specimen 
which he used was evidently far from pure*. This formula, 
which is the most natural to adopt for representing the 
deviations from lineality in a case of this kind, had been 
previously employed to a limited extent by others for the 
variation of resistance with temperature; but it had not pre- 
viously been proved to be suitable to represent this particular 
phenomenon over so extended a range. 

The work of the Committee of the British Association in 
1874 was mainly confined to investigating the changes of 
zero of a Siemens pyrometer when heated in an ordinary fire 
to moderately elevated temperatures. Finding that the pyro- 
meter did not satisfy the fundamental criterion of givinc 
always the same indication at the same temperature, it did. 
not seem worth while to pursue the method further, and the 
question remained in abeyance for several years. In the 
meantime great advances were made in the theory and 
practice of electrical measurement, so that when I com- 
menced to investigate the subject at the Cavendish Labo- 
ratory, the home of the electrical standards, in 1885, I was 
able to carry out the electrical measurements in a more 
satisfactory manner, and to avoid many of the sources of 
error existing in previous work. The results of my investi- 
gations were communicated to the Boyal Society in June 
1886, and were published, with additions, in the ; Philoso- 
phical Transactions ' of the following year. Owing to a 
personal accident, no complete abstract of this paper as a 
whole was ever published ; and as the paper in its original 
form is somewhat long and inaccessible, many of the points 
it contained have since been overlooked. The greater part of 
the paper was occupied with the discussion of methods and 
observations with air-thermometers ; but it may not be amiss 
at the present time to give a summary of the main conclusions 

* It may be remarked that the sign of this coefficient for platinum and 
palladium is wrongly quoted in Wiedemann, JSlectricitat, vol. i. p. 525. 

Prof. H. L. Callendnr on Platinum Thermometry. 195 

■which it contained, so far as they relate to the subject of 
platinum thermometry. 

(1) It was shown that a platinum resistance-thermometer, 
if sufficiently protected from strain and contamination, was 
practically free from changes of zero over a range of 0° to 
1200° C, and satisfied the fundamental criterion of giving 
always the same indication at the same temperature. 

(2) It was proposed to use the platinum thermometer as a 
secondary standard, the temperature pt on the platinum scale 
being defined by the formula 

^ = 100(R-E°)/(R / -E°), .... (1) 

in which the letters R, R°, R / si and for the observed resist- 
ances at the temperatures pt, 0°, and 100° C. respectively. 

(3) By comparing the values of ' pt deduced from different 
pairs of specimens of platinum wires, wound side by side and 
heated together in such a manner as to be always at the same 
temperature, it was shown that different wires agreed very 
closely in giving the same value of any temperature pt on the 
platinum scale, although differing considerably in the values 
of their temperature-coefficients. (See below, p. 209.) 

(4) A direct comparison was made between the platinum 
scale and the scale of the air-thermometer by means. of several 
different instruments, in which the coil of platinum wire was 
enclosed inside the bulb of the air-thermometer itself, and so 
arranged as to be always at the same mean temperature as the 
mass of air under observation. As the result of this com- 
parison, it was shown that the small deviations of the platinum 
scale from the temperature t by air-thermometer could be 
represented by the simple difference-formula 

D = t-pt = d(tl\00-l)tll00, .... (2) 

with a probable error of less than 1° C. over the range 0° to 
650° 0. 

(5) It was inferred from the comparisons of different 
specimens of wire referred to in (3) (which comparisons were 
independent of all the various sources of error affecting the 
air-thermometer, and could not have been in error by so much 
as a tenth of a degree) that the simple parabolic formula did 
not in all cases represent the small residual differences between 
the wires. 

(6) It was shown by the direct comparison of other typical 
metals and alloys with platinum, that the temperature-variation 
of the resistance of metals and alloys in general could probably 
be represented by the same type of formula over a consider- 
able range with nearly the same order of accuracy as in the 

196 Prof. H. L. Callendar on Platinum Thermometry. 

case of platinum. But, that the formula did not represent 
singularities due to change of state or structure, such as those 
occurring in the case of iron at the critical temperature, or in 
the case of tin at the point of fusion. 

This paper attracted very little attention until the results 
were confirmed by the independent observations of Griffiths**, 
who in 1890 applied the platinum thermometer to the deter- 
mination of certain boiling- and freezing-points, and to the 
testing of mercury thermometers of limited scale. The 
results of this work appeared at first to disagree materially 
with the difference-formula already quoted, the discrepancy 
amounting to between 6° and 7° at 440° C. After his work 
had been communicated to the Royal Society a direct com- 
parison was made with one of my thermometers in his appa- 
ratus ; and the discrepancy was traced to the assumption by 
Griffiths of RegnaiuYs value 448°'38 C. for the boiling-point 
of sulphur. We therefore undertook a joint redetermination 
of this point with great care, employing for the purpose one 
of my original air-thermometers which had been used in the 
experiments of 1886. The results of this determination were 
communicated to the Royal Society in December 1890, and 
brought the observations of Griffiths into complete harmony 
with my own and with the most accurate work of previous 
observers on the other boiling- and freezing-points in question. 
The agreement between his thermometers when reduced by 
the difference-formula (2), employing for each instrument the 
appropriate value for the difference-coefficient d, was in fact 
closer than I had previously obtained with platinum wires 
from different sources. But the agreement served only to 
confirm the convenience of the method of reduction by 
means of the Sulphur Boiling-Point (S.B.P.) which we 
proposed in that paper -"-. 

Proposed Standard Notation and Nomenclature. 

It will be convenient at this stage, before proceeding to 
discuss the results of later work, to explain in detail the 
notation and phraseology which I have found to be useful in 
connexion with platinum thermometry. This notation has 
already in part been adopted by the majority of workers 
in the platinum scale, and it would be a great saving in time 
and space if some standard system of the kind could be gene- 
rally recognized. In devising the notation special attention 
has been paid to the limitations of the commercial typewriter, 
as the majority of communications to scientific societies at 

* Phil. Trans, clxxxii. (1891), A, pp. 4S-72. 
f Ibid. t.c. pp. 119-157. 

Prof. H. L. Callendar on Platinum Thermometry. 197 

Ihe present time are required to be typewritten. It is for 
this reason desirable to avoid, wherever possible, the use of 
Greek letters and subscript diacritics and indices. 

The Fundamental Interval. — The denominator, B/ — R°, in 
formula (1) for the platinum temperature pt, represents the 
change of resistance of the thermometer between 0° and 
100° C, and is called the fundamental interval of the thermo- 
meter, in accordance with ordinary usage. It is convenient, 
as suggested in a previous communication, to adjust the 
resistance of each thermometer, and to measure it in terms 
of a unit such that the fundamental interval is approxi- 
mately 100. The reading of the instrument will then give 
directly the value of pt at any temperature, subject only to a 
small percentage correction for the error of adjustment of the 
fundamental interval. 

The Fundamental Coefficient. — The mean value of the tem- 
perature-coefficient of the change of resistance between 0° 
and 100° C. is called the fundamental coefficient of the wire, 
and is denoted by the letter c. The value of c is given by the 
expression (B/ - R°)/100 R°. The value of this coefficient is 
not necessary for calculating or reducing platinum tempe- 
ratures, but it is useful for identifying the wire and as giving 
an indication of its probable purity. 

The Fundamental Zero. — The reciprocal of the fundamental 
coefficient c is called the fundamental zero of the scale of the 
thermometer, and. is denoted by the symbol pt°, so that 
pt°=l/c. The fundamental zero, taken with the negative 
sign, represents the temperature on the scale of the instrument 
itself at which its resistance would vanish. It does not 
necessarily possess any physical meaning, but it is often more 
convenient to use than the fundamental coefficient (e. g. y 
Phil. Trans. A, 1887, p. 225). It may be remarked that, 
if the resistance has been accurately adjusted so that the 
fundamental interval is 100 units, R°, the resistance at 0° C, 
will be numerically equal to pt°. 

The Difference Formula. — It is convenient to write the 
formula for the difference between t and pt in the form already 
given (2), as the product of three factors, d x (£/100— -1) x t/\ 00, 
rather than in the form involving the square of £/100, which I 
originally gave, and which has always been quoted. Owing 
to the form in which it was originally cast, I find that most 
observers have acquired the habit of working the formula in 
the following manner. First find the square of f/100, 
then subtract £/100, writing the figures down on paper, and 
finally multiply the difference by the difference-coefficient d 
with the aid of a slide-rule. It is very much easier to work 
Phil. Mag. S. 5. Vol. 47. No. 285. Feb. 1899. P 

198 Prof. H. L. Callendar on Platinum Thermometry. 

the formula as the product of three factors, because the sub- 
traction (7/100 — 1) can be safely performed by mental 
arithmetic. The whole process can then be performed by 
one application of the slide-rule, instead of two, and it is 
unnecessary to write down any intermediate steps on paper. 

The Parabolic Function. — It is convenient to have an 
abbreviation for the parabolic function of t vanishing at 
0° and 100°, which occurs so frequently in questions of 
thermometry. I have found the abbreviation p(t) both sug- 
gestive and useful for this purpose. The formula may then 
be written in the abbreviated shape, t=pt-\-dxp(t). 

The "S.B.P." Method of Reduction.— -Assuming the differ- 
ence-formula, the value of the difference-coefficient d may be 
determined by observing the resistance B", and calculating 
the corresponding value of the platinum temperature pt" . at 
some secondary fixed point t n , the temperature of which is 
known on the scale of the gas-thermometer. The boiling- 
point of sulphur (S.B.P.) is generally the most convenient 
to use, and has been widely adopted for this purpose. As- 
suming that this point is chosen for the purpose, and that the 
height of the barometer at the time is 760 + A millims., the 
corresponding temperature is given by the formula 

t" =444-53 + -082 h, 

provided that h is small, and the corresponding value of the 
parabolic function by the formula 

p(t") = 15*32 + -0065 h, 

whence d=(t"-pt // )[p(t // ). 

With the best apparatus it is possible to attain an order of 
accuracy of about 0*1 per cent, in the value of d obtained by 
this method, at least in the case of thermometers which are 
not used at temperatures above 500°. At higher temperatures 
the exact application of the formula would be more open to 
question, and it may be doubted whether the value of the 
difference-coefficient would remain constant to so small a 
fraction of itself. 

Other Secondary Fixed Points. — For very accurate work 
between 0° and 100° C. it might be preferable to use a value 
of d determined at 50° C, either by direct comparison with 
an air-thermometer or by comparison with a standard platinum 
thermometer. The latter comparison would be much the 
easier and more accurate. Although the most careful com- 
parisons have hitherto failed to show that the value of d 
obtained by assuming the S.B.P. does not give correct results 

Prof. H. L. Callendar on Platinum Thermometry. 199 

between 0° and 100° C, it is quite possible that this might not 
always be the case. 

For work at low temperatures it would be preferable, from 
every point of view, to make use of the boiling-point of oxygen 
as the secondary fixed point. There appears to be a very 
general consensus of opinion that the temperature of liquid 
oxygen boiling under a pressure of 760 mm. is — 182°*5 C, 
on the scale of the constant-volume hydrogen or helium 
thermometer*. It is quite possible that, as in the case of 
water and most other liquids, the temperature of the boiling 
liquid would be different from that of the condensing vapour 
at the same pressure ; but the boiling liquid is the most con- 
venient to employ, and it appears that its temperature is 
steady to two or three tenths of a degree, and reproducible by 
different observers to a similar order of accuracy. I have 
found it convenient for purposes of distinction to employ the 
symbol d° to denote the value of d deduced from the boiling- 
point of oxygen, and the symbol d" to denote that deduced 
from the boiling-point of sulphur. The formulae for the 
pressure correction in the case of oxygen are approximately 

t= -182-5 + -020 A; p(t) = 5-16--00093A. 

The Resistance Formula. — I have shown in the paper 
already referred to that the adoption of the parabolic differ- 
ence-formula for the relation between pt and t is equivalent 
to assuming for the resistance- variation the formula 

R,/B° = l + at + bt* (3) 

The values of the coefficients a and b are found in terms of c 
and d, or vice versa, by means of the relations 

a=c {l + d/100), b=-cd/10fl00. 

Graphic Method of Reduction. — The quickest and most 
generally convenient method of reducing platinum tempe- 
ratures to the air- scale is to plot the difference t— pt in terms 
of t as abscissa, and to deduce graphically the curve of differ- 
ence in terms of pt as abscissa, as described and illustrated 
in my original paper. This method is particularly suitable at 
temperatures up to 500° C, as the difference over this range 
is relatively small and accurately known. It is also very con- 
venient if a large number of determinations are to be made 
with a single instrument. It is not so convenient in the case 
of a number of different instruments with different coefficients, 

* The experimental evidence for this number is not quite satisfactory, 
owing to differences in the atmospheric pressure and impurities in the 
oxygen. It must be understood that the adoption of this value is 
provisional and subject to correction. 


200 Prof. H. L. Callendar on Platinum Thermometry. 

each of which is used for a comparatively limited number of 
determinations. In such a case the trouble of drawing the 
separate curves, with sufficient care to be of use, would more 
than counterbalance the advantage to be gained by the 

Hey cock and Neville's Method. — In order to avoid this 
difficulty Messrs. Heycock and Neville, in their classical 
researches at high temperatures * devised an ingenious modi- 
fication of procedure, which has given very good results 
in their hands, but is not quite identical with the simple 
difference-formula. They described a difference-curve in the 
usual manner, giving the value of the difference in terms of 
ft as abscissa for a standard value rf = l*50 of the difference- 
coefficient. The appropriate values of d were determined in 
the case of each pyrometer by the S.B.P. method. In re- 
ducing the observations for any given values of pt and <7, the 
value of the difference corresponding to pt was taken from 
the curve for J = l*50, and was then multiplied by the factor 
d/1'50 and added to pt. This method is very expeditious 
and convenient, and gives results which are in practical 
agreement with the pure difference-formula, provided that, 
as was almost invariably the case in their observations, the 
values of d do not differ materially from the average 1*50. 
If, however, the pure difference-formula is correct, the method 
could not be applied in the case of values of d differing con- 
siderably from the average. The difference between the 
methods cannot be simply expressed in terms of either pt or t 
for considerable variations in the value of d. But for a small 
variation Sd in the value of d in the vicinity of the normal 
value, it is easy to show that the difference St between 
the true value of t as given by the difference-formula 
t—pt=.dp(t), and the value found by the method of Heycock 
and Neville, is approximately 


Neglecting the variation of d entirely, the error would be 
B r t=Sd(dt/dpt)p{t). 

For example, at £ = 1000°, p(t)=90, (d£/dpf) = l-40, we 
should find for a variation of d from 1*50 to I'60, the values 
& = 3°-8 (H. & N.), and 8't =12°'8 (variation neglected). 

This is an extreme case. In the observations of Heycock 
and Neville, the values found for the coefficient d seldom 

* Trans. Chem. Soc. Feb. 1895, p. 162. 

t The value of dt/dpt at any point is readily found by differentiating 1 
the difference-formula (2), dpt/dt=l-(t/50- ljd/100. 

Prof. H. L. Callendar on Platinum Thermometry. 201 

varied so much as # 04 on either side of the mean, in the case 
of their standard wire. It is, moreover, quite possible that 
these variations may have been partly due to fortuitous differ- 
ences at the S.B.P. and at the fixed points, in which case it 
is probable that the Heycock and Neville method of reduction 
would lead to more consistent results than the pure difference- 
formula, because it does not allow full weight to the apparent 
variations of d as determined by the S.B.P. observations. 
It is clearly necessary, as Heycock and Neville have shown, 
and as the above calculation would indicate, to take some 
account of the small variations of d, at least in the case of 
pyrometers in constant use at high temperatures. The method 
of Heycock and Neville appears to be a very convenient and 
practical way of doing this, provided that the variations of d 
are small. It must also be observed that, although the indi- 
vidual reductions by their method may differ by as much as 
1° or 2° at 1000° from the application of the pure difference- 
formula, the average results for the normal value of d will be 
in exact agreement with it. 

Difference- Formula in Terms of pt. — In discussing the 
variation of resistance as a function of the temperature, it is 
most natural and convenient to express the results in terms of 
the temperature t on the scale of the air-thermometer by 
means of the parabolic formula already given. This formula 
has the advantage of leading to simple relations between the 
temperature-coefficients ; and it also appears to represent the 
general phenomenon of the resistance-variation of metals 
over a wide range of temperature with greater accuracy than 
any other equally convenient formula. When, however, it 
is simply a question of finding the temperature from the 
observed value of the resistance^ or from the observed reading 
of a platinum thermometer, over a comparatively limited 
range, it is equally natural, and in some respects more con- 
venientj to have a formula which gives t directly in terms of 
pt or R. This method of expression was originally adopted 
by Griffiths, who expressed the results of the calibration of 
his thermometers by means of a formula of the type 

t—pt = apt + bpt 2 + cpt d + dpt 4 . ...(G) 

The introduction of the third and fourth powers of pt in 
this equation was due to the assumption of RegnauhVs value 
for the boiling-point of sulphur. If we make a correction 
for this, the observations can be very fairly represented by a 
parabolic formula of the type already given, namely, 

t-pt=dXpt/100-l)pt/l0Q=d / p(pty . . (I) 

202 Prof. H. L. Callendar on Platinum Thermometry. 

This formula is so simple and convenient, and agrees so 
closely over moderate ranges of temperature with the ordinary 
difference-formula, as to be well worth discussion. I have 
been in the habit of using it myself for a number of years in 
approximate reductions at moderate temperatures, more par- 
ticularly in steam-engine and conductivity experiments, in 
which for other reasons a high degree of accuracy is not 
required. It has also been recently suggested by Dickson 
(Phil. Mag., Dec. 1897), though his suggestion is coupled 
with a protest against platinum temperatures. 

The value of the difference- coefficient oV in this formula 
may be determined as usual by reference to the boiling-point 
of sulphur, or it may be deduced approximately from the 
value of the ordinary difference-coefficient d by means of the 

d'=d/(l--077d), or d=d'/{l + '077d'). 

If this value is chosen for 
ence-formulae will of course 

the coefficient, the two dii 

at 0°, 100 c 

and 445° C, 
The order of 

but will differ slightly at other temperatures, 
agreement between the formulae is shown at various points of 
the scale by the annexed table, in which t represents the 
temperature given by the ordinary formula t—pt=l'50p(t), 
and t' the temperature calculated by formula (4) for the same 
value of pt, choosing the value d'= 1*695, to make the two 
formulae agree at the S.B.P. 

Table I. 
Comparison of Difference-Formulae, (2) & (4). 


-4° -5 






300° ! 

t-v ... 






+22° -9 




t-t' ... 

It will be observed that the difference is reasonably small 
between the limits — 200° and + 600°, but that it becomes 
considerable at high temperatures. A much closer agreement 
may be readily obtained over small ranges of temperature by 
choosing a suitable value of d' . The two formulae become 
practically indistinguishable between 0° and 100°, for in- 
stance, if we make d f = d. For steam-engine work I generally 
selected the value of d f to make the formulae agree at 200° C. 

Prof. H. L. Callendar on Platinum Thermometry. 203 

For work at low temperatures, it would be most conveuient to 
select the boiling-point of oxygen for the determination of 
either difference-coefficient. The two formulae are so similar 
that they cannot be distinguished with certainty over a 
moderate range of temperature. But if the values of the differ- 
ence-coefficients are calculated from the S.B.P., the balance 
of evidence appears to be in favour of the original formula (2). 
Formula (4) appears to give differences which are too large 
between 0° and 100° C; and it does not agree nearly so 
well as (2) with my own air- thermometer observations over 
the range 0° to 650° C. It appears also from the work of 
Heycock and Neville to give results which are too low at high 
temperatures as compared with those of other observers. 

It is obvious, from the similarity of form, that the differ- 
ence-formula (4) in terms of pt corresponds, as in the case of 
formula (2), to a parabolic relation between the temperature 
and resistance, of the type 

^-^H-a'R/^ + Z/CR/R^^a'XR/^-lJ+^^R/R -!) 2 . (5) 

When R = 0, t = - t° = -(<*"-&"). Also V= b", and 

The values of the fundamental coefficient c, and of the fun- 
damental zero pt°, are of course the same on either formula, 
provided that they are calculated from observations at 0° and 
100° C, but not, if they are calculated from observations 
outside that range. The values of the coefficients a" and h" 
are given in terms of d' ', and either pt°, or c, by the relations 

a" =pt° (1 - d'/lOO) = ( 1 - d7 100) /c, and d! = 10,0006V. 

Formulae of this general type, but expressed in a slightly 
different shape, have been used by Holborn and Wien for 
their observations at low temperatures, and recently by 
Dickson for reducing the results of Fleming and other 
observers. But they do not employ the platinum scale or 
the difference-formula. 

Maximum and Minimum Values of the Resistance and 
Temperature, — It may be of interest to remark that the dif- 
ference-formulae (2) and (4) lead to maximum or minimum 
values of pt and t respectively, which are always the same 
for the same value of d, but lie in general outside the range 
of possible extrapolation. In the case of formula (2), the 
resistance reaches a maximum at a temperature t= —a/'lb — 
(5000/d) (l + d/100). The maximum values of pt and R are 
given in terms of d and c by the equations 

pt (max.) = (1 + d/100) t/2 = (2500/rf) (1 + d/lOO)*, 

R/R° (max.) =l+pt (max.)//rf?=l+ (2500c/d) (l+^/lOO) 2 . 

204 Prof. H. L. Gallendar on Platinum Thermometry. 

Similarly in the case of the difference-formula (4) in terms 
of pt, the maximum or minimum value of t is given in terms 
of d' by the equation 

*(max.) = (l-dyiOOXp*/2= -(2500/d') (l-tf'/lOO) 2 . 

Dickson's Formula. — In a recent number of this Journal 
(Phil. Mag., Dec. 1897) Mr. Dickson has proposed the formula 

(R+ a y= P (t+b) (6) 

He objects to the usual formula (3) on the grounds, (1) 
that it leads to a maximum value of the resistance in the 
case of platinum at a temperature of about £ = 3250° C, and 
(2) that any given value of the resistance corresponds to two 
temperatures. He asserts that " both of these statements 
indicate physical conditions which we have no reason to sup- 
pose exist/' In support of contention (1), he adduces a rough 
observation of Holborn and Wien * to the effect that the 

* Wied. Ann. Oct. 1895 ; p. 386. Mr. Dickson and some other writers 
appear to attach, too much weight to these observations of Messrs. Holborn 
and Wien. So far as they go, they afford a very fair confirmation of the 
fundamental principles of platinum- thermometry at high temperatures ; 
but the experiments themselves were of an incidental character, and 
were made with somewhat unsuitable apparatus. Only two samples of 
wire were tested, and the resistances employed were too small for 
accurate measurement. The wires were heated in a badly-conducting 
inutile and were insulated by capillary tubes of porcelain or similar 
material. The temperature of the wire under test was assumed to be the 
mean of the temperatures indicated by two thermo-junctions at its 
extremities ; but the authors state that " the distribution of temperature 
in the furnace was very irregular." The resistance was measured by a 
modification of the potentiometer method, and no attempt was made to 
eliminate residual thermoelectric effects. Under these conditions the 
observations showed that the resistance was not permanently changed by 
exposure to a temperature of 1600° C, at least within the limits of 
accuracy of the resistance measurements. It is quite easy, however, by 
electric heating as in the " nieldonreter," to verify the difference-formula 
at high temperatures, with less risk of strain or contamination or bad 
insulation. (See Petavel, Phil. Trans. A (1898), p. 501.) 

The two series of observations (excluding the series in which the tube 
of the muffle cracked, and the thermocouples and wire were so con- 
taminated with silicon and furnace-gases as to render the observations 
valueless) overlapped from 1050° to 1250° C, and showed differences 
between the two wires varying from 10° to 45° at these temperatures, 
the errors of individual observations in either series being about 10° to 
15°. It must be remembered, however, that the two wires were of 
different sizes and resistances ; they were heated in different furnaces ; 
they were insulated with different materials; and their temperatures 
were deduced from different thermocouples. Taking these facts into 
consideration, it is remarkable that the observed agreement should be so 
close. The observations at the highest temperatures in both cases, with 
the furnaces full blast and under the most favourable conditions for 
securing uniformity of temperature throughout the length of the wire, are 
in very close agreement with the difference-formula (2), assuming d=l-75. 

The second specimen was also tested at lower temperatures, but the 

Prof. H. L. Callendar on Platinum Thermometry. 205 

resistance of one of their wires had already nearly reached 
6R° at a temperature of about l(U0 n C, whereas the maximum 
calculated resistance in the case of one of my wires (with a coeffi- 
cient c= '00340) was only (r576R°. He omits to notice that 
the result depends on the coefficients of the wire. 

The wire used by Holborn and Wien had a fundamental 
coefficient c = '00380, and the highest value of the resistance 
actually observed was not 6R° as suggested, but B/R° — 5'53, 
at a temperature £ = 1610° C, deduced from thermo-j unctions 
at each end. If we assume d=l'70 as a probable value of 
the difference-coefficient for their wire, the difference-formula 
(2) would give, 

at £=1610°, D = 414°, ^=1196°, whence R/R° = 5'54. 

It w T ould be absurd to attach much weight to so rough an 
observation, but it will be seen that, so far as it goes, the 
result is consistent with the usual formula, and does not bear 
out Mr. Dickson's contention. A more important defect in 
arguments (1) and (2) lies in the fact that maximum and 
minimum values of the resistance are known to occur in 
the case of manganin and bismuth within the experimental 
range, and that such cases can be at least approximately 
represented by a formula of the type (3), but cannot be 
represented by a formula of the type (6). 

As shown by Table I. above, the formula proposed by 
Dickson agrees fairly well with formula (3), in the special 
case of platinum, through a considerable range. Rut the 
case of platinum is exceptional. If we attempt to apply a 
formula of Dickson's type to the case of other metals, we are 
met by practical difficulties of a serious character, and are 
driven to conclude that the claim that it is 6l more represen- 
tative of the connexion between temperature and resistance 
than any formula hitherto proposed/' cannot be maintained. 

observations are somewhat inconsistent, and lead to values of d which 
are rather large and variable, ranging from 3:7 to 2-0. These variations 
are probably due to errors of observation or reduction. This is shown 
by the work of Mr. Tory (B.A. Report, 1897), who made a direct com- 
parison between the Pt — PtRh thermocouple and the platinum-thermo- 
meter by a much more accurate method than that of Holborn and Wien. 
He found the parabolic difference-formula for the platinum thermometer 
to be in very fair agreement between 100° and 800° C. with the previous 
series of observations of Holborn and Wien on this thermocouple (Wied. 
Ann. 1892), and there can be little doubt that the discrepancies shown 
by their later tests were due chiefly to the many obvious defects of the 
method. For a more detailed criticism of these observations, the reader 
should refer to a letter by Griffiths in ' Nature,' Feb. 27th, 1896. It is 
sufficient to state here that the conclusions which these observers drew 
from their experiments are not justified by the observations themselves. 

206 Prof. H. L. Callendar on Platinum Thermometry. 

If, for instance, we take the observations of Fleming on very 
pure iron between 0° and 200° C, and calculate a formula of 
the Holborn and Wien, or Dickson, type to represent them, 
we arrive at a curve similar to that shown in fig. 1 (p. 193). 
(The values of the specific resistance of Fleming's wire are 
reduced, for the sake of comparison, to the value R = 10,000 
at 0° C.) This curve agrees very closely with that of Morris 
and other observers between 0° and 200° C. The peculiarities 
of the curve beyond this range are not due to errors in the 
data, but to the unsuitable nature of the formula. A similar 
result would be obtained in the case of iron by employing 
any other sufficiently accurate data. It will be observed that 
the formula leads to a maximum value of the temperature 
£ = 334°, and makes the resistance vanish at —197°. Below 
334° there are two values of the resistance for each value of 
the temperature, and the value of dR/d£ at 334° is infinite, 
both of which conditions are at present unknown in the case 
of any metal, and are certainly not true in the case of 
iron. If, instead of taking the value observed at + 196°*1 C, 
we take the value obtained at the O.B.P. to calculate the 
formula, we should find a better agreement with observation 
at low temperatures, but the disagreement at higher tempe- 
ratures would be greater. 

If, on the other hand, we take the same observations, namely, 
c = '00625, and R/B,°= 2*372 when * = 196°'l, and calculate a 
difference-formula of the type (2) corresponding to (3), we 
find a 7 =-12-5, a = -005467, b = -000,007825. The points 
marked in fig. 1 are calculated from this formula, and are 
seen to be in practical agreement with the observations of 
Morris up to 800°. As this formula stands the test of extra- 
polation so much better than that of Holborn and Wien or 
Dickson, we are justified in regarding it as being probably 
more representative of the connexion between resistance and 

Advantages of the Difference- Formula. — Mr. Dickson's ob- 
jections to the platinum scale and to the difference-formula 
appear to result from want of familiarity with the practical 
use of the instrument. But as his remarks on this subject 
are calculated to mislead others, it may be well to explain 
briefly the advantages of the method, which was originally 
devised with the object of saving the labour of reduction 
involved in the use of ordinary empirical formulae, and of 
rendering the results of observations with different instruments 
directly and simply comparable. 

(1) In the first place, a properly constructed and adjusted 
platinum thermometer reads directly in degrees of temperature 

Prof. H. L. Callendar on Platinum Thermometry. 207 

on the platinum scale, just like a mercury thermometer, or 
any other instrument intended for practical use. The quantity 
directly observed is not the resistance in ohms, but the tem- 
perature on the platinum scale, either pt, or j)t+pt°. The 
advantage of this method is that the indications of different 
instruments become directly comparable, and that the values 
of pt for different wires agree very closely. If this method 
is not adopted, the resistances in ohms of different instruments 
at different temperatures form a series of meaningless figures, 
which cannot be interpreted without troublesome reductions. 

(2) The second advantage of the difference-formula lies in 
the fact that the difference is small, more especially at mode- 
rate temperatures, and can be at once obtained from a curve 
or a table, or calculated on a small slide-rule, without the 
necessity of minute accuracy of interpolation or calculation. 
In many cases, owing to the smallness of the difference 
between the scales, the results of a series of observations 
can be worked out entirely in terms of the platinum scale, 
and no reduction need be made until the end of the series. 
For instance, in an elaborate series of experiments on the 
variation of the specific heat of water between 0° and 100° C, 
on which I have been recently engaged, by a method de- 
scribed in the Brit. Assoc. "Report, 1897, all the observations 
are worked out in terms of the platinum scale, and the re- 
duction to the air-scale can be performed by the aid of the 
difference-formula in half an hour at the end of the whole 
series. As all the readings of temperature have to be taken 
and corrected to the ten-thousandth part of a degree, and as 
the whole series comprises about 100,000 observations, it is 
clear that the labour involved in Mr. Dickson's method of 
reduction would have been quite prohibitive. It is only by 
the general introduction of the method of small corrections 
that such work becomes practicable. 

On the Method of Least Squares. — There appears to be a 
widespread tendency among non-mathematical observers to 
regard with almost superstitious reverence the value of results 
obtained by the method of least squares. This reverence in 
many cases is entirely misplaced, and the method itself, as 
commonly applied, very often leads to erroneous results. For 
instance, in a series of observations extending over a con- 
siderable range of temperature, it would be incorrect to attach 
equal weight to all the results, because all the sources of 
error increase considerably as we depart further from the 
fixed points of the scale. In a series of air-thermometer 
observations, the fixed points themselves stand in quite a 
different category to the remainder of the observations. The 

208 Prof. H. L. Callendar on Platinum Thermometry. 

temperature is accurately known by definition, and is not 
dependent on uncertain errors of the instrument. It is a 
mistake, therefore, in reducing a series of observations of this 
kind, to put all the observations, including the fixed points, 
on the same footing, and then apply the method of least 
squares, as Mr. Dickson has applied it in his reduction of the 
results of various observers with platinum thermometers. For 
instance, in order to make his formula fit my observations at 
higher temperatures, he is compelled to admit an error of no 
less than o, 80 on the fundamental interval itself, which is 
quite out of the question, the probable error of observation on 
this interval being of the order of o, 01 only. The correct 
way of treating the observations would be to calculate the 
values at the fixed points separately, and to use the remainder 
of the observations for calculating the difference-coefficient. 
Even here the graphic method is preferable to that of least 
squares, because it is not easy to decide on the appropriate 
weights to be attached to the different observations. Cor- 
recting the method of calculation in this manner, we should 
find a series of differences between my observations and 
Dickson's formula, of the order shown in Table I. It would 
be at once obvious that the deA^ations from (6) were of a 
systematic type, and that it did not represent the results of 
this series of observations so well as that which I proposed. 
The deviations shown in Dickson's own table are of a syste- 
matic character ; but they would have been larger if he had 
treated the fixed points correctly. 

Limitations of the Difference-Formula. — The observations 
of Messrs. Haycock and Neville at high temperatures may be 
taken as showing that the simple parabolic difference-formula, 
in which the value of d is determined by means of the S.B.P. 
method, gives very satisfactory results, in spite of the severe 
extrapolation to which it is thus subjected, provided that the 
wire employed is of pure and uniform quality. If, however, 
the S.B.P. method of reduction is applied in the case of impure 
wires at high temperatures, it may lead to differences which 
are larger than the original differences in the values of pt 
before reduction. For instance, I made a number of pyro- 
meters some years ago with a sample of wdre having the 
coefficients c = '00320, d // = l'7b. My observations on the 
freezing-points of silver and gold (Phil. Mag., Feb. 1892) 
were made with some of these pyrometers. All these instru- 
ments gave very consistent results, but they could not be 
brought into exact agreement with those constructed of purer 
wire by the simple S.B.P. method of reduction, employ- 
ing either difference-formula (2) or (4). This is not at all 

Prof. H. L. Callendar on Platinum Thermometry. 209 

surprising when we consider the very large difference in the 
fundamental coefficient c, which is approximately '00390 in 
the case of the purest obtainable wire. The remarkable fact 
is that, as stated in my original paper (see above, p. 195), the 
values of pt for such different specimens of wire should show 
so close an agreement through so wide a range. The differ- 
ence in the fundamental coefficients in this extreme case is about 
20 per cent.; but the values oipt for the two wires differ by only 
4° at the S.B.P., and this difference, instead of increasing in 
proportion to the square of the temperature, remains of the 
same order, or nearly so, at the freezing-points of silver and 
gold. Thus the wire c = '00320 gave p£ = 830° at the 
Ag. F.P., but I shortly afterwards obtained with a specimen 
of very pure wire (c = '003897), the value pt = 835° for the 
same point. Messrs. Reycoek and Neville, using the same 
pure wire, have confirmed this value. They also find for the 
F.P. of gold, with different instruments, constructed of the 
same wire, the average value p£ = 905 o, 8. I did not test this 
point with the pure wire, but the value found by Messrs. 
Heycock and Neville may be compared with the value 
^ = 902°'3 (Phil. Mag., Feb. 1892), which I found at the 
Mint with one of the old instruments. 

From these and other comparisons of the platinum scales of 
different wires, it appears likely that the deviation of the 
impure wire from the parabolic curve is generally of this 
nature. As shown by the comparison curves in my original 
paper, the deviation follows approximately the parabolic law 
up to 400°, beyond that point the curves tend to become 
parallel, and at higher temperatures they often show a 
tendency to approach each other again. The application of 
the S.B.P. method of reduction to impure wires at high 
temperatures will therefore give results which are too high, 
because the value of d is calculated from the S.B.P., where 
the difference between the wires is nearly a maximum. Thus, 
taking the values of d from the S.B.P. for the two specimens 
of wire above quoted, we find, calculating the values of t for 
the Ag.F.P., and Au.F.P. from the data, 

Impure wire, c = '00320, d= 1*751 ; 

Ag. F.P., £ = 98P6 ; Au. F.P., £ = 1092'0. 
Pure wire, c = '00390, d = l'520 ; 

Ag. F.P., £ = 960'7 : Au. F.P., £=1060'7. 

The results for the impure wire obtained by the S.B.P. method 
of reduction are not so high as those found by Bams with a 
Pt-Ptlr thermo-element, which he compared with an air- 
thermometer up to 1050°. There can be little doubt. 

210 Prof. H. L. Callendar on Platinum Thermometry. 

however, that they are too high, and that the results given 
"by the pure wire are the more probable. The latter are 
approximately a mean between the values of Violle 954°, and 
Holborn and Wien 971°, and may be taken, in the pre- 
sent state of the science of high-temperature measurement, 
to be at least as probable as any other values, in spite of 
the extrapolation from 445°, by which they are obtained. 

The extrapolation is not really so unreasonable as many 
observers seem to think. The parabolic formula for resistance 
variation has been verified for a great variety of cases, through 
a very wide range, and with much greater accuracy than in 
the case of many so-called laws of nature. For instance, a 
similar formula, proposed by Tait and Avernarius, is often 
regarded as the law of the thermocouple, but the deviations 
of thermocouples from this law are far wider than those of the 
most impure platinum thermometer. If we take a Pt-PtRh 
thermocouple, and apply the S.B.P. method of reduction in 
the same manner as in the case of a platinum thermometer, 
taking the data, t= 100°, e=650 microvolts; £ = 445°, e = 
2630 mv. ; we should find d= -7*4. At *=1000°C., e=9550 
mv., the temperature on the scale of the thermocouple is 
^ = 1470°. The temperature calculated by the parabolic 
formula is £ = 804°. Whence it will be seen that the devia- 
tion from the formula is about ten times as great as in the 
case of a very impure platinum wire. A cubic formula was 
employed by Holborn and Wien to represent their observations 
at hio-h temperatures with this thermocouple, but even this 
formula differs by more than 20° from their observations at 
150° C. It is, moreover, so unsatisfactory for extrapolation 
that they preferred to adopt a rectilinear formula for deducing 
temperatures above 1200° C. 

There are, however, more serious objections to the adoption 
of the thermocouple, except to a limited extent, as a secondary 
standard: — '(1) The scale of the thermocouple is seriously 
affected, as shown by the observations of Holborn and Wien 
and Barus at high temperatures, and of Fleming at low 
temperatures, by variations in the quality of the platinum 
wire and in the composition of the alloy. (2) The sen- 
sitiveness of the Pt — PtEh thermocouple at moderate 
temperatures is too small to permit of the attainment of 
the order of accuracy generally required in standard work. 
(3) No satisfactory method has yet been devised in the case 
of the thermocouple for eliminating residual thermal effects 
in other parts of the circuit, which materially limit* the 

* My present assistants, Prof. A. W. Porter, B.Sc, and Mr. N. Eumor- 
fopoulos, B.Sc, whose work on Emissivity and Thermal Conductivity has 
already in part been published in this Journal, employed this thermo- 

Prof. H. L. Callendar on Platinum Thermometry. 211 

attainable accuracy. In the case of the platinum thermo- 
meter these effects are relatively much smaller, owing to the 
large change of resistance with temperature, and can be 
completely eliminated in a very simple manner. 

Ag. F.P. Method of Reduction for Impure Wires. — The 
simplest method of reduction for such wires at high tempera- 
tures, would be to take the Ag. F.P. as a secondary fixed 
point instead of the S.B.P. for the determination of the 
difference-coefficient d. This would in general lead to a very 
close agreement at temperatures between 800° and 1200°C, 
but would leave residual errors of 3° or 4° at temperatures in 
the neighbourhood of the S.B.P. To obtain a continuous 
formula giving results consistent to within less than 1° 
throughout the range, it would be necessary to adopt the 
method which I suggested in my last communication (Phil. 
Mag., Feb. 1892), assuming d to be a linear function of the 
temperature of the form a-\-bt, and calculating the values of 
a and b to make the instrument agree with the pure wire at 
both the S.B.P. and the Ag. F.P., taking the latter as 960°'7. 
We should find for the wire (c = "00320) above quoted, d= 
1-580 at the Ag. F.P. If we apply this value at the Au. F.P., 
we should find £ = 1063 o, 0. But if we employ the second 
method, and calculate a linear formula for d to make the 
results agree throughout the scale, taking d= 1*751 at the 
S.B.P., we obtain d=a+ fa = T898 — 000331*. Hence 
the appropriate value of d to use at the Au. F.P. would be 
d= 1;547, giving for the Au. F.P. £ = 1060°-0, which is in 
closer agreement with the value 1060 o, 7 given by the pure 
wire. This method has also the advantage that it gives 
practically perfect agreement at the S.B.P., and at all points 
between 0° and 1000°. In the case of the mercury thermo- 
meter,, or the thermocouple, a similar cubic formula is required 
to give an equally good agreement between 0° and 200° 0. 

In the original paper in which the suggestion was made, I 

couple very extensively in their investigations. They inform me that 
they were compelled to abandon the method shortly before my appoint- 
ment, because in spite of every precaution which their experience could 
suggest they found it impossible, owing to these residual thermal effects, 
to effect a sufficiently accurate calibration of the Pt-PtRh thermo- 
couple at temperatures between 0° and 100° C. The substitution of 
baser metals such as iron and german-silver at low temperatures would 
no doubt partly meet this difficulty, but would involve the abandonment 
of the wide range and constancy and uniformity of scale characteristic of 
the platinum metals, which are qualifications so essential for a standard. 
We conclude on these grounds that the application of this thermo- 
couple is limited to high temperatures, and tnat the contention that it is 
preferable to the platinum thermometer as a secondary standard cannot 
be maintained. 

212 Prof. H. L. Callendar on Platinum Thermometry. 

assumed tentatively a much lower yalue £=945° for the Ag. 
F.P., giving a result £=1037° for the Au. F.P., which 
naturally does not agree with the results of subsequent work. 
These results have since been misquoted in a manner which 
has the effect of suggesting that the platinum thermometer 
gives very capricious results at high temperatures. Holborn 
and Wien, for instance, quote my value 981°'6 for the Ag. 
F.P., obtained with the impure wire by the S.B.P. method 
of reduction, and at the same time quote the value 1037° 
for the Au. F.P., which was obtained by assuming the value 
945° for the Ag. F.P. Comparing these with the values 
obtained by Heycock and Neville with the pure w T ire, one 
might naturally conclude, in the absence of information as to 
the manner in which the two results were calculated, that 
different wires gave very inconsistent results. The truth is, on 
the contrary, that very different wires agree with remarkable 
uniformity in giving approximately the same platinum-seal e, 
and that they also give consistent values of t provided that the 
reduction is effected in a consistent manner. But, although 
it is evident that this method may be made to give consistent 
results in the case of impure wires, it is in all cases preferable 
to use pure wire of uniform quality. If, forinstancej a pyro- 
meter gives a value of c less than '0035, or a value of d 
greater than 1*70, it would be safer to reject it, although it 
may possibly give very consistent results. Values of d greater 
than 2*00 at the S.B.P. sometimes occur, but may generally 
be taken as implying that the wire is contaminated. Such 
instruments as a rule deteriorate rapidly, and do not give 
consistent results at high temperatures. 

The Difference- Formula at Loiv Temperatures. — The suita- 
bility of the Platinum thermometer as an instrument for low- 
temperature research is shown by the work of Dewar and 
Fleming, and Olszewski. It has also been adopted by Holborn 
and Wien, in spite of their original prejudice against the 
instrument. The first verification of the platinum scale at 
very low temperatures was given by Dewar and Fleming, 
whose researches by this method are the most extensive and 
important. They found that two different specimens of wire 
with fundamental coefficients c = "00353, and c = "00367 
respectively, agreed very closely in giving the same values of 
the platinum temperature down to —220°. The values of 
the difference-coefficients for these wires, calculated by 
assuming t= — 182°*5 for the boiling-point of liquid oxygen, 
are d = 2'15 * and d = 2'72, respectively. The first of these 
refers to the particular wire which Dewar and Fleming 
selected as their standard. 

* See below, p. 219, middle, and footnote. 

Prof. H. L. Callendar on Platinum Thermometry. 213 

As an illustration of the method of reduction by the differ- 
ence-formula, it may be of interest to reproduce a table 
exhibiting in detail the complete calculation of such a table of 
reduction for the standard wire employed by Dewar and 
Fleming. We select for this purpose the following corrected 
data, taken from their paper in the Phil. Mag., July 1895, 
p. 100. 

Thermometer in Melting Ice, B,° = 34059, t = Q°Q. 

„ Steam at 760 mm., R / = 4'2034, £ = 100° C. 

Liquid Oxygen, . R" = 0'9473, t- -182°'5C. 
From these data we deduce : — 

Fundamental Interval, R' -P° = 1-0975. 
Fundamental Coefficient, (R'-B, o )/100R° = -003533. 
Fundamental Zero, y=l/c=283°'00. 
In Liquid Oxygen, 

p t=-W6°'7, f=-182°-5, B = t-pt = U°'2. 
Difference-Coefficient, d = I>/p(t) = U'2/5'lQ = 2'75. 
Difference-Formula, D = *-/?* = 2"75(*/100-l)*/100. 
To find the difference-formula in terms of pt, we have 

Difference-Coefficient, d' = D/p(pt) = 14'2/5'84 = 2'43. 
Pt Difference-Formula, D' = *'-7rt = 2-43(p*/100- 1)^/100. 

As a verification we may take the observation in solid C0 2 
and ether, assuming Regnault's value t= — 78 0, 2 for the true 

Difference-Formula (D) gives, t-pt=2'75 x 1-39 = 3°-82. 
„ „ (D ; ) „ /'-^ = 2-43xl-49 = 3°-62. 

The observed value of ptis given as — 81 0, 9. Thus the 
two formulae give, (D) f=s-78°-l, and (D') ^=-78°'3, re- 

The following Table shows the comparison of the formulae 
for every ten degrees throughout the range. The first three 
columns contain the whole work of the calculation for formula 
(D') . The second column contains the values of D' calculated 
by the aid of a small slide-rule. These when added to the 
values of pt in the first column, give the values of t shown in 
the third column. The fourth column contains the correspond- 
ing-values of the difference in t for 1° pt, obtained by differen- 
tiating the difference-formula. These are written down by 
the method of differences. The fifth column contains the 
difference t — t' between the values of / deduced by the two 
formulae. The sixth contains the values of t by formula (D) ; 
and the seventh is added for comparison with the table given 
by Dickson (Phil. Mag., June 1898, p. 527). 

Phil. Mag. S. 5. Vol. 47. No. 285. Feb. 1899. Q 

214 Prof. H. L. Callendar on Platinum. Thermometry. 

Table II. — Table of Reduction for Dewar and Fleming's 
Standard Platinum Thermometer. 

pt (°C). 






+ 99-85 

+ 100 

+ 100 


+ 50 







+ o 



- 9-70 

+ 0-20 
- 9-51 

- 10 


- 9-73 


- 20 







- 30 







- 40 

+ 1-36 






- 50 

+ 1-82 






- 60 







- 70 







- 80 







- 90 





- 85-62 










+ 560 







+ 641 







+ 7-28 







+ 8-14 







+ 912 



+ 0-22 




+ 10-1 







+ 11-2 







+ 123 







+ 134 



| +0-05 




+ 14-6 



! -002 






+ 15-8 











4 18-4 










| -0-43 







1 -0-58 







! -0-73 























The above table affords a good illustration of the point 
already mentioned, that the results obtained from the two 
differ en ce-fornmlse (D) and (D') agree so closely over a limited 
range, as in the present case, that it is often quite immaterial 
which of the two is used for purposes of reduction. The 
largest difference over the experimental range in the present 
instance is only 0°"3, which is less than many of the errors of 
observation, except at the fixed points and under the most 
favourable conditions. In comparing the two formula the 
following expression for the difference between them is 
occasionally useful : — 

D-D / =^-^'=^D(2^ + D-100)/10,000+(^'-l)D / . 

Prof. H. L. Callendar on Platinum Thermometry. 215 

It is generally sufficient to put D = D' on the right-hand side 
of this formula, so that if either is known the difference 
between them may be determined with considerable accuracy. 

It will be observed that the table of reduction given by 
Dickson agrees very closely w r ith either of the difference- 
formulae. But, on the whole, most closely with (D). If 
Dickson had calculated his formula from the same data it 
would have given results identical with (D'). By giving 
equal weight, however, to all the observations, without regard 
to steadiness of temperature or probable accuracy, he is com- 
pelled, as in the previous instance, to admit an error of o, 35 
in the fundamental interval itself, which is quite impossible. 
Except at these points the probable error of his reduction is 
not of vital importance ; on the contrary, the general agree- 
ment with (D) is so close that it is difficult to see on what 
grounds he can regard the latter as being either incorrect or 

For practical purposes a table of this kind is not convenient 
owing to the continual necessity for interpolation. A graphic 
chart in w T hich t is plotted directly against pt is objectionable, 
because it does not admit of sufficient accuracy unless it is 
plotted on an unwieldy scale. The difference- curve avoids 
this difficulty, and is much to be preferred for laboratory work. 
But for occasional reduction it is so easy to calculate the 
difference directly from the formula that it is not worth while 
to take the trouble to plot a curve. 

Reduction of Olszewski's Observations. — The observations 
of Olszewski on the critical pressure and temperature and 
boiling-point of hydrogen, described in the Phil. Mag, for 
July 1895, were made with a platinum thermometer of '001 
inch w T ire wound on a mica frame in the usual manner. He 
graduated this thermometer by direct comparison with a 
constant-volume hydrogen thermometer at the lowest tem- 
peratures which he could obtain by means of liquid oxygen 
boiling under diminished pressure. The lower temperatures, 
observed with the thermometer immersed in temporarily 
liquefied hydrogen, were deduced from the observed resist- 
ances by rectilinear extrapolation, assuming that the resistance 
of the platinum thermometer continued to decrease, as the 
temperature fell, at the same rate as over the lowest tempe- 
rature interval, —182*5 to —208*5, included in the range of 
the comparison with the hydrogen thermometer. It is pos- 
sible that, at these low temperatures, the resistance of platinum 
does not continue to follow the usual formula, but it may be 
interesting to give a reduction of his observations by the 
difference method for the sake of uniformity of expression. 


216 Prof. H. L. Callendar on Platinum Thermometry. 

We select for this purpose the following data : — 
Thermometer in Melting Ice, 
E/R°= 1-000, *=0°C. 
Thermometer in solid C0 2 at 760 mm., 

R/R° = -800, ^=-78-2° C. 
Thermometer in Liquid 2 at 760 mm., 
R/R° = -523, *=-182°-5 C. 
From these we deduce the folio wing values of the coefficients : — 

a=-002515, b = --000,000,53, c = '002462, rf=2-13, 

As a verification we have the observation R/R° = *453, at 

£=-208°'5 C. This gives pt= -222% D=13°'7 ; which 
agrees with the value given by the difference-formula calcu- 
lated from the three higher points. 

The following Table owes the reduction of the observations 
taken with this thermometer in partially liquid hydrogen. 

Table III. — Reduction of Olszewski's Observations 
in Boiling Hydrogen. 





t (° C). 

t Olszewski. 






















The effect of this change in the method of reduction is to 
make the temperature of the boiling-point of hydrogen nearly 
one degree higher than the value given by Olszewski. If we 
employ instead the difference-formula in terms of pt, we should 
find c' = -002472, pt° = 404°'5, d'=l'85. This formula leads 
to the values given in the column headed t' , which are a little 

The value found by Dewar for liquid hydrogen (Proc. R. S. 
Dec. 16,1898) is much higher, namely t= — 238° # 8 at one 
atmo, and — 239°*6* at l/30th atmo. The difference may 
possibly be due to the superheating of the liquid, or, more 
probably, to some singularity in the behaviour of his thermo- 
meter at this point (see below, p. 218) . 

Observations of Holborn and Wien (Wied. Ann. lix. 1896). 
— Holborn and Wien made a direct comparison between the 

* Values calculated from observed resistances by formula (2). 


Prof. H. L. Oallendar on Platinum Thermometry. 217 

hvdrogen and platinum thermometers, adopting my method 
of enclosing the spiral inside the bulb of the air-thermometer. 
The majority of their observations were taken while the tem- 
perature of the instrument was slowly rising. This method 
of procedure is very simple, but it is open to the objection 
that the mean temperature of the spiral is not necessarily 
the same as that of the gas enclosed, especially when, as in 
their apparatus, the spiral is asymmetrically situated in an 
asymmetrical bulb. If we take their observations in melting- 
ice, in solid C0 2 , and in liquid air, which are probably in this 
respect the most reliable, and calculate a difference-formula 
in terms of pt, we shall find c' = *003621, d , = l'69. Calcu- 
lating the values of t' by this formula, we find that all the 
rest of their observations make the temperature of the plati- 
num spiral on the average 1° higher than that of the gas. 
This might be expected, as the temperature was not steady, 
and the warmer gas would settle at the top of the bulb, the 
spiral itself being also a source of heat. 

If we take their own formula, and calculate the equivalent 
difference-formula, we find c' = '003610, d'= 1*79. This agrees 
fairly well with the values found above, as they appear also 
to have attached greater weight to the observations in C0 2 
and liquid air. But, if we take the formula calculated by 
Dickson (Phil. Mag. Dec. 1897), who attaches equal weight 
to all their observations, we find c' = '003527, d'=2'43. The 
excessive difference in the values of the coefficients deduced 
by this assumption is an index of the inconsistency of the 
observations themselves*. 

Behaviour of Pure Wire at Low Temperatures, — In the 
case of ordinary platinum wire, with a coefficient c = '0035 or 
less, the effect of the curvature at low temperatures of the 
t, R, curve, as represented by the positive value of the dif- 
ference-coefficient d, is to make the resistance diminish more 
rapidly as the temperature falls, and tend to vanish at a point 
nearer to the absolute zero than the fundamental zero of 
the wire itself. When, however, the value of pt° is numeri- 
cally less than 273°, the effect of this curvature would be to 
make the resistance vanish at some temperature higher than 
the absolute zero. If, therefore, we may assume that the 
resistance ought not to vanish before the absolute zero, we 
should expect to find a singular point, or a change in sign of 
the difference-coefficient, at low temperatures. If this were 
the case, it would seriously invalidate the difference-formula 
method of reduction, at least at low temperatures, and as 

* Contrast the close agreement of Dickson's reduction in the case of 
Fleming's observations. 

218 Prof. H. L. Callendar on Platinum Thermometry. 

applied to wires for which pt° was numerically less than 
273°. When, therefore, I succeeded in obtaining in 1892 a 
very pure specimen of wire, with the coefficient c = '00389, 
pt° =257°, I quite expected to find it behave like iron and 
tin, w T ith the opposite curvature to the impure platinum, and 
a negative value for the coefficient d. On testing it at the 
S.B.P. and also at the Ag.F.P. I found, on the contrary, that 
it gave a value d=+l*50, and that its scale agreed very 
closely with that of all the other platinum wires I had tested, 
at least at temperatures above 0° 0. I sent a specimen to 
Prof. Fleming shortly afterwards and he used it as the 

mo- thermometer P 9 " in his researches on the thermo- 


electric properties of metals at low temperatures. The test 
of this wire is given by Fleming in the Phil. Mag. July 
189 5, p. 101, from which the following details are extracted: — 

c = -003885,jrt° = 257°-4. C0 2 B.P., pt= -81°-3. 

Assuming *=-182°-5 at the O.B.P., we have ^=+2'10, 
which gives £=— 78 c, 4 for the temperature of solid C0 2 . 
The value of the difference-coefficient, so far from vanishing 
or changing sign, appears to be actually greater at very low 
temperatures. According to this formula, the resistance of 
the wire tends to vanish at a temperature t° = — 240 o, 2, cor- 
responding to 2)t°= — -257°-4. It seems not unlikely, however, 
according to the observations of Do war, that the resistance, 
instead of completely vanishing at this temperature, which is 
close to the boiling-point of hydrogen, ceases to diminish 
rapidly just before reaching this point, and remains at a small 
but nearly constant value, about 2 per cent, of its value at 0° C. 
Application of the Difference-Formula to the case of other 
Metals. — The application of the difference-formula is not 
limited to the case of platinum. It affords a very convenient 
method of reduction of observations on the resistance-varia- 
tion of other metals. I employed it for this purpose in the 
comparison of platinum and iron wires *, as a means of veri- 
fying the suitability of the parabolic formula for the expres- 
sion of variation of resistance with temperature. Thus, if the 
symbol ft stands for the temperature by an iron- wire thermo- 
meter, defined by formula (1), in exactly the same manner 
as the platinum temperature, and if d and d' stand for the 
difference-coefficients of platinum and iron respectively, as- 
suming that both wires are at the same temperature t, we 
have clearly the relation 

ft-pt={d-d f )Xp(t). 

* PM1. Trans. A. 1887, p. 227. 

Prof. H. L. Callendar on Platinum Thermometry. 219 

As an illustration of the convenience of this method of re- 
duction a table is appended giving the values of the constants 
at low temperatures for the specimens tested by Dewar and 
Fleming. The data assumed in each case are (1) the value 
of the fundamental coefficient c given in the first column, and 
(2) the value of the temperature of the O.B.P. on the scale 
of each particular metal, calculated from the observed re- 
sistance by formula (1), and given in the third column. The 
value of the difference-coefficient d° for each metal as deduced 
from the O.B.P. is found at once by the relation 


The sign of this coefficient indicates the direction of the cur- 
vature of the temperature-resistance curve, and its magnitude 
is approximately proportional to the average relative curvature 
over the experimental range. 

The values of the coefficients a and b, given in the last two 
columns, are readily calculated from those of c and d by 
means of the relations already given (p. 199). These co- 
efficients refer to the equivalent resistance - formula ( 3 ), 
and are useful for calculating the specific resistance at any 

In comparing the values of d°, given in this table, with 
those deduced from observations at higher temperatures, it 
will be noticed that they are in most cases algebraically 
greater, the difference amounting to nearly 30 per cent, in 
many cases between the values deduced from the O.B.P. and 
the S.B.P. respectively. It is possible that this indicates a 
general departure from the exact parabola requiring further 
experiments for its elucidation. It would be unsafe, however, 
to infer from the results of the present investigation that this 
is always the case, because, owing to the construction of the 
coils with silk and ebonite insulation, it was impossible to 
test the wires directly in sulphur, and they could not be 
annealed after winding at a higher temperature than 200°. 
It is well known that annealing produces a marked effect on 
the form of the curve and on the value of d*. It is also stated 
in the paper that trouble was experienced from thermoelectric 
disturbances, owing to the use of thick copper leads 4 mm. 
in diameter. Such effects cannot be satisfactorily eliminated 
except by the employment of a special method of compensa- 

* With reference to this point it is interesting to remark that Messrs. 
Heycock and jNeville with one of their perfectly annealed pyrometers of 
pure wire, for which c = '00387, e7= T497, found the value pt=— 80 o, 3, 
t=— 78 0, 2C, for the C0 2 B.P. This would perhaps iudicate that the 
larger values of d were due to imperfect annealing. 

220 Prof. H. L. Callendar on Platinum Thermometry. 


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P3 "S e 



c -3 



















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■g p^" 1 * 


















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1— 1 

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Prof. H. L. Callendar on Platinum Thermometry. 221 

tion, which will be described in a subsequent communication. 
The general result of any residual thermal effects which may 
be present is to produce a change in the apparent value of d, 
since the thermo-E.M.F. follows approximately a parabolic 
formula. It is possible, for this reason, to obtain consistent 
and accurate measurements of temperature with a platinum 
thermometer in spite of large thermal effects, but the value of 
d would be very considerably affected. 

On the " Vanishing Temperature" — There appears to be a 
very general consensus of opinion, based chiefly on the par- 
ticular series of experiments which are under discussion, that 
the resistance of all pure metals ought to vanish, and does 
tend to vanish at a temperature which is no other than the 
absolute zero. If, hoAvever, there is any virtue in the para- 
bolic method of reduction, it is quite obvious, on reference to 
the column headed " Vanishing Temperature " in the above 
table, that the resistance "tends to vanish" in the case of 
most of the common metals at a much higher temperature. 
The vanishing temperature f is the value of t deduced from 
the fundamental zero pt° in each case by means of the 
difference-formula, employing the value of d° given in the 
table. The most remarkable metals in this respect are pure 
copper and iron, which tend to become perfect conductors at 
a temperature of —223° approximately, a point which is now 
well within the experimental range. These are followed at a 
very short interval by aluminium, nickel, and magnesium. 
In the case of copper and iron special experiments were made 
at a temperature as low as —206° 0., at which point the 
rate of decrease of resistance showed little, if any, sign of 
diminution. The exact value of the vanishing temperature 
in each case is necessarily somewhat uncertain owing to the 
necessity of extrapolation, and also on account of possible 
uncertainties in the data ; but there can be no doubt that the 
conclusion derived from the formula represents, at least ap- 
proximately, a genuine physical fact. Whether or no the 
resistance does actually vanish at some such temperature may 
well be open to doubt. It would require very accurate ob- 
servations to determine such a point satisfactorily, as the ex- 
perimental difficulties are considerable in measuring so small 
a resistance under such conditions. It is more probable that 
there is a singular point on the curve, similar to that occur- 
ring in the case of iron at the critical temperature, at which 
it ceases to be magnetic. It is also likely that the change 
would not be sudden, but gradual, and that indications of the 
approaching singularity would be obtained a few degrees 
above the point in question. Below this point it is even pos- 

222 Messrs. Rosa and Smith on a Calorimetr 


sible that the resistance might not tend to vanish, but, as in 
the case apparently of bismuth, might increase with further 
fall of temperature. It has been suggested that at very low 
temperatures all metals might become magnetic. It is very 
probable that the change of electrical structure here indicated 
would be accompanied by remarkable changes in the magnetic 
properties. These are some of the points which experiment 
will probably decide in the near future. The only experi- 
mental verification at present available is the observation of 
Dewar in the case of platinum No. 3 when immersed in 
boiling hydrogen at —240° C, that the resistance after at- 
taining a very low value apparently refused to diminish 
further, in spite of a considerable lowering of the pressure. 
It would be extremely interesting to repeat this observation 
with specially constructed thermometers of copper or iron, 
which ought to show the effect in a more striking manner 
and at a higher temperature. 

My thanks are due to Messrs. E. H. Griffiths, C. T. Hey- 
cock, and F. H. Neville, and to Prof. A. W. Porter and 
Mr. N. Eumorfopoulos, for their kind assistance in revising 
and correcting the proofs of this article. 

X1Y. A Calorimetric Determination of Energy Dissipated in 
Condensers. By Edward B. Rosa and Arthur W. Smith*. 

IN a former paper {supra, p. 19} we gave the results of mea- 
surements by means of a wattmeter of the energy dissipated 
in condensers when they were subjected to an alternating 
electromotive force. The results were such that we desired 
to confirm them by a totally independent method : and, in 
addition, to measure the energy dissipated in some paraffined- 
paper condensers which showed so small a loss that with the 
coils at our disposal the Resonance Method, employed success- 
fully on beeswax and rosin condensers, would not give 
sufficiently accurate values. We therefore constructed a 
special calorimeter for the purpose of measuring the total 
quantity of heat produced in the condensers, which represents 
the total energy dissipated. 

Fig. 1 gives an external view of the calorimeter, and fig. 2 
a vertical section. The calorimeter proper, A, is the inner of 
three concentric boxes, and is 33 cm. long, 30 cm. deep, and 
10 cm. in breadth. It has a copper lining, a, and a copper 
jacket, b, and is protected by the two exterior boxes from 
fluctuations of temperature without. The general principle 
of the calorimeter is (1) to prevent any loss or gain of heat 
* Communicated by the Authors. 

Determination of Energy Dissipated in Condensers. 223 

through its walls, and (2) to carry away and measure all heat 
generated within by a stream of water To effect the first con- 
dition two concentric copper walls (the lining and the jacket) 
are maintained as nearly as possible at the same temperature. 
This, of course, will reduce the flow of heat through the inter- 
vening wooden wall to a minimum, and make the " cooling 
correction " small, if not zero. 

(i r> 


To Eliminate the Cooling and Capacity Corrections. 

In order to ascertain any difference of temperature between 
the copper walls a and b, a differential air-thermometer is 
used. Each air-chamber of this differential thermometer 
consists of a copper pipe about 4 metres long and 4 millim. 
internal diameter, one coiled about and soldered to the lining, 

224 Messrs. Rosa and. Smith on a Calorimetric 

and the other coiled ahont and soldered to the jacket 


end of each pipe is closed and the other connected to one end 
of the U-tube, G, shown on the outside of the calorimeter in 
fig. 1. The U-tube, which we call the gauge, contains kero- 
sene oil, and serves to indicate any difference of temperature 
between the two copper walls. The zero-mark is fixed after 

Fig. 2. 
E F 

maintaining the whole calorimeter at a constant temperature 
for some hours. In order to keep the gauge reading sensibly 
zero, and thus keep the two copper walls very closely at the 
same temperature, a coil of wire through which an electric 
current of any desired strength can be passed is wound about 
the jacket in the space B. And in order to make the regulation 
more perfect a second coil is wound about the second box in 

Determination of Energy Dissipated in Condensers. 225 

the space C, so as to maintain the temperature of this space 
nearly constant. The temperature of the chamber A is 
usually kept a little higher than the external temperature, so 
that no cooling is required ; and by varying the currents in 
the two heating-coils the temperature in B can be made to 
follow that in A so closely that the gauge-readings are always 
small, and their algebraic sum during any experiment zero. 
This eliminates all correction for radiation. In rare cases 
when the temperature of the room has risen considerably, we 
have found it necessary to hang a wet cloth about the box to 
prevent the temperature of C rising above that of B and A. 
We intend to coil a small copper pipe in C so that a stream of 
cool water may be sent through it, and then no difficulty will 
be encountered in the hottest weather. 

In addition to the gauge four thermometers (fig. 1) indicate 
the temperatures of A, a, b, and C: that is, A' shows the tem- 
perature of the air in the calorimeter chamber A ; a! has its 
bulb in a pocket of the lining a, and hence indicates the tem- 
perature of the wall a; V similarly extends down into a 
pocket of the copper jacket b, and shows its temperature. 
Finally, C 7 gives the temperature of the outer air-space C. 
A' is an accurate thermometer reading from i0° to 25° C, 
graduated to 0°*01 and read to o> 001 C. If A 7 shows the 
temperature to be constant during the whole period of an 
experiment, or the same for a considerable time near the end 
of an experiment that it was at the beginning, then there will 
be no correction for heat absorbed or given up by the appa- 
ratus. With both the " cooling correction " and the capacity 
correction eliminated, it remains to carry away and measure 
the entire heat generated by a condenser in A, or by any other 
source of heat within the calorimeter. 

2. Carrying away and Measuring the, Heat. 

In order to carry away the heat generated a stream of 
water, which enters at 1 (figs. 1 and 3), is made to flow 
through a coiled copper pipe (fig. 3), where it absorbs heat, 
and then leaves the calorimeter at 0. In order to increase its 
absorbing capacity the pipe is soldered to a sheet of copper, 
L L, both pipe and copper being painted black. Three such 
sheets, each with 4 metres of pipe attached, are joined 
together and placed side by side in the chamber A, the con- 
densers being slipped in between them. The rate of absorption 
of heat depends upon the difference of temperature between 
the absorbers and the air surrounding them. If a large 


Messrs. Rosa and Smith on a Calorimetric 

amount of heat is to be brought away, the water is made to 
enter at a low temperature and to flow rapidly through the 
absorbers. If a smaller quantity of heat is to be absorbed 
and carried away, the entering water will be warmer, and its 
gain in temperature correspondingly less. By varying the 
temperature of the water and its rate of flow, the rate of 
absorption can be varied between wide limits, and kept very 

Fig. 3. 

nicely at any desired point. In practice the thermometer A' 
is the guide in regulating the temperature of the entering 
water. If the temperature of A begins to rise (A 7 , as already 
stated, can be read to one-thousandth of a degree), the entering 
water is slightly cooled; if to fall, it is slightly warmed ; the 
rate of flow of water, after being once adjusted for a given 
experiment, is maintained constant. 

In order to measure the quantity of heat thus carried 
away, the thermometers E and F are inserted in two small 
reservoirs, M and N, which stand in the wooden wall of the 
calorimeter between the two copper surfaces. The thermo- 
meter E indicates the temperature of the water just as it 
enters the chamber A, and the thermometer F gives its tem- 
perature as it leaves. The difference of temperature multiplied 

Determination of Energy Dissipated in Condensers. 227 

by the mass of water per second gives the rate of absorption 
and removal of heat. The thermometers are accurately gra- 
duated and read to hundredths of a degree. The gain in 
temperature is several degrees, and may be ten or twenty 
degrees by increasing the quantity of heat generated or 
reducing the rate of flow of water. Hence the accuracy of 
the determination of the quantity of heat absorbed is sufficient 
for most purposes. The chief error is ordinarily due to 
changes in the temperature of the apparatus itself and its 
contents. By running the experiment several hours, however, 
and keeping it as nearly as possible at a constant temperature, 
this uncertainty is greatly reduced and the error made 
negligibly small. 

The water flows into the calorimeter from a reservoir about 
a metre above, this height furnishing the necessary pressure. 
The temperature of the entering water is regulated by adding 
warm or cold water to the reservoir, and the rate of flow of 
the water is regulated by an adjustable valve. The water is 
collected in a litre flask, the time of each litre being recorded. 

3. Test of the Calorimeter. 

Table I. shows the result of one of the tests made upon the 
calorimeter. A current of electricity passed through a coil 
of wire within the chamber A, the electromotive force being- 
measured by a carefully calibrated Weston voltmeter, and 
the current by a Kelvin balance. The experiment continued 
for a little more than four hours, while nine litres of water 
passed through the calorimeter. The rate of absorption of 
heat was nearly, but not quite, constant, the temperature as 
indicated by A! having varied slightly. The final tempe- 
rature was practically the same as that at the beginning, being 
slightly higher if anything. The average for the nine litres 
is 12*37 watts absorbed and carried away by the water, while 
the electrical measurements give 12*34 watts. By continuing 
the experiment longer and introducing greater refinements in 
the measurement of the current and electromotive force, a 
greater degree of accuracy could undoubtedly be attained. 
But this and other tests showed clearly that for our present 
purposes the calorimeter was abundantly accurate, and we 
proceeded to put some condensers into it and measure the heat 


Messrs. Rosa and Smith on a Calorimetric 

Table I. 

Test of the Calorimeter. 



Period in 



for each 

h. m. s. 

of water. 

1 53 00 

2 19 45 


2 46 50 


3 13 55 


3 41 15 


4 09 10 


4 37 10 


5 04 10 


5 32 15 


6 01 00 




ture of 


(«) (/) 

Average ' Increase 
tempera- in tempe- 

15 57 

I ture of 

| the out- 

i i going 


rat lire of 
I each 
I of water. 






(/)X1000. (g)^(c) 










watts = 
calories X J. 




Electromotive force =20*0 volts. 

Current=0-617 ampere. 

Watts (from electrical measurements) 

20-0x0-617= 12-34. 

4. The Experiments. 

Table II. (p. 230) give? the results of six experiments with 
the same beeswax and rosin condensers which were employed 
in onr work by the resonance method. In each experiment a 
preliminary run, not included in the table, allows the condensers 
and calorimeter to come to a constant temperature. Column 
(a) gives the numbers of the condensers in each case, they all 
being joined in parallel to the same electromotive force. 
Column (b) gives the time of the beginning of each litre of 
water; column (c) the duration of each litre or 1000 grammes 
of water; column (d) the average temperature of the ingoing- 
water as found from readings of the thermometer E, taken 
regularly every five minutes, and column (e) the same for the 
outgoing wafer; (/) then shows the increase of temperature. 
Column (g) gives the number of calories of heat carried away 
by each 1000 grammes of water, and column (Ji), which is the 
number in (g) divided by the corresponding number of seconds 
recorded in (c), is the rate of absorption of heat. Column (i) 
gives the number of watts to which this is equivalent, taking 

Determination of Energy dissipated in Condensers. .229 

J, the mechanical equivalent of heat, to be 41,972,000 ergs. 
This is the value derived from Rowland's and Griffiths's work, 
assuming the specific heat of water at an average temperature 
of 20° C. to be unity. Column (j) gives the frequency. 
Sometimes this was estimated from the average frequency of 
the dynamo supplying the lines of the Middle town lighting- 
circuits at the time ; and in other cases it was determined by 
measuring the speed of a small synchronous motor. The 
electromotive force (k) was measured with an electrometer, the 
current (I) with a Siemens dynamometer. Column (??) gives 
the values of cos cf> of the expression power = EI cos <£. Care 
was taken in every instance to avoid the presence of upper 
harmonics, in some cases using a resonance-coil to quench the 
harmonics as well as increase the voltage on the condenser. 
Column (o) gives the per cent, loss, 1007T cot </>, and column 
(p) the net efficiency. This relative loss, tt cot <$>, has been 
proved* to be the ratio of w to W, where w is the energy 
dissipated per half-period, and W is the energy stored in the 
condenser at each charge. 1— 7rcot<£, the net efficiency, is 

Energy stored — Energy lost 
Energy stored 

5. Beeswax and Rosin Condensers. 

The first experiment, with condensers Nos. 3, 4, and 7, 
showed a net efficiency of 93*39 per cent., or a loss of 6*61 
per cent. The temperature of the condensers was not deter- 
mined ; but from the fact that the dielectric was softened and 
the quantity of heat generated was more than in any suc- 
ceeding experiment, we feel sure that it was considerably 
above 40° 0. Six condensers were then placed in the calori- 
meter, joined in three pairs. Nos. 1 and 5 gave no sound 
when joined to an alternating E.M.F. of 1000 or more volts, 
and we called it the " best pair/' In Nos. 3 and 4 vibrations 
were distinctly felt when the fingers were placed in contact 
with them, while the condensers gave a clear musical note 
and on the higher voltages a hissing sound ; this we called 
the " poorest pair." Nos. 8 and 9 were intermediate. 

These six condensers were first of all joined in parallel and 
connected to a low-frequency circuit of 1520 volts and 26 
periods per second. Care was taken to exclude upper har- 
monics. The experiment continued over three hours after 
the temperature of calorimeter and condensers had become 

* See our paper, Phil. Mag. Jan. 1893. 
Phil. Mag. S. 5. Vol. 47. No. 285. Feb. 1899. R 


Messrs. Rosa and Smith on a Calorimetric 

Table II. — Measurements of Energy dissipated 









No. of 

Period in 



in tempe- 

Total heat 



per second. 

seconds for 



rature of 



1000 grams 

ture of the 

ture of the 

1000 grams 



of water. 



of water. 





h m 8 
9 36 00 




3, 4, 

10 02 00 








10 25 00 







10 51 30 








6 34 35 



7 12 30 







7 49 15 







8 26 30 



22 08 




9 05 10 







9 41 40 








2 56 30 

3&4 S 

3 27 10 







3 55 45 







4 24 00 







4 52 15 







5 21 20 








7 10 40 

7 28 13 







1 & 5. 

7 56 40 







8 25 47 







8 55 50 







9 26 20 








2 38 30 

3 11 30 







1 &5. 

3 44 20 


20 90 





4 17 20 







4 50 20 


20 68 







8 37 20 



9 15 20 







9 54 15 







Determination of Energy Dissipated in Condensers. 231 

in Beeswax and Rosin Condensers. 












calories X J. 







Per cent 
loss = 

7TCOt(pX 100. 

Efficiency = 


(A) X 4-1972. 






100 -(o). 


















































232 Messrs. Rosa and Smith on a Calorimetric 

constant by a preliminary run of several hours. The tem- 
perature of the calorimeter as indicated by the thermometer 
A' rose gradually for an hour, and hence the heat absorbed 
was less than the average. During the last hour the tempe- 
rature was reduced by quickening the rate of flow and cooling 
the entering water, so that the temperature was substantially 
the same at the end as at the beginning. The per cent, of 
loss is 4*84 at an average temperature of the condensers of 
30° C. The voltage employed on this low-frequency test was 
much higher than for any other experiment, and yet there 
was no evidence of brush-discharge or appreciable leakage- 

In the third, experiment the same condensers were subjected 
to a high-frequency electromotive force at 30° C, and the loss 
found to be 5*06 per cent., that is, slightly greater than before. 
Hence for a given voltage the energy dissipated, per period 
would be slightly greater, and the energy dissipated per second 
more than five times as much as for the low frequency. 

In the fourth experiment only the " best pair " of con- 
densers was used, and with a slightly higher voltage the 
temperature of the condensers rose to 40° C. Here the los3 
was found to be 8*37 per cent., nearly as much as the maximum 
value found by the resonance method. 

The fifth experiment, with the " best pair," was made 
some days later at 30° C, and the percentage loss came out 
6*35 per cent., that is greater than the average of the six. This 
was unexpected, as well as the last result, which showed a loss 
for condensers 3 and 4, the " poorest pair," of 4*82 per cent., 
which was less than the average. These results were then 
confirmed by an independent method, showing conclusively 
that the so-called " poorest pair " had the smallest loss ; not, 
of course, because it emitted a distinct sound and hissed on 
high voltages, but in spite of that. The chief loss is doubtless 
due to some cause quite independent of the singing and 
hissing, and happens to be smaller where it would naturally 
be expected to be larger. 

Thus we have confirmed by these calorimetric measure- 
ments the large values of the losses which we found by the 
resonance method in beeswax and rosin condensers, and also 
the existence of a well-marked maximum as the temperature 
rises, beyond which the loss decreases considerably. It is an 
interesting fact that the residual charges of these condensers 
are very large, that they increase with the temperature up to 
40° C, and then decrease as the temperature is carried 
higher. That is, the maximum point for the residual charge is 
the same as for the energy loss. 

Determination of Energy dissipated in Condensers. 233 

6. Paraffined- Paper Condensers. 

The second lot of condensers used were commercial paraffined- 
paper condensers made by the Stanley Electric Co. A finished 
condenser is a solid, slab about 25 x 30 cm. and 2 cm. thick, 
thus having a volume of 1500 c. c, and is enclosed in a tight 
tin case, the lead-wires coming out through ebonite bushings. 
Nos. 1 to 4 of our condensers have a capacity of about 
1*7 microfarad each ; Nos. 5 to 10, which were purchased 
about a year later, have a capacity of about 3*2 microfarads 
each. The condensers of the second lot are made of paper 
about -0038 cm. thick, two sheets being placed together in 
each stratum. This we learned by dissecting some which we 
had broken down. It ought to be stated, however, that 
while the condensers are guaranteed by the makers to stand 
500 volts alternating electromotive force, we have repeatedly 
subjected them to 1000 to 2000 volts, and in some cases for 
several hours at a time. Nos. 9 and 10 were upon one 
occasion maintained at 2250 (effective) volts, at a frequency 
of 130, for over an hour, and showed no signs of being over- 
taxed. We have, however, broken several at voltages between 
1000 and 2000. The paper of the first lot of condensers is 
thicker, but as we have never broken one of this lot we 
cannot state its thickness. From the fact that the capacity 
of each of these is about 60 per cent, as great as that of the 
others, while their volumes are substantially the same, we 
conclude that the thickness of the paper is about "0048 cm., 
supposing there are, as in the other, two sheets together in 
every stratum of the condensers. 

In Table III. are given the results of seven separate expe- 
riments with Stanley condensers, which were made at intervals 
during the past three months. The frequency in every case 
except experiment 4 was estimated from the average frequency 
of the two dynamos of the Middletown lighting circuits. 
Experiments 1 and 3 were made when the faster dynamo was 
supplying the lines, the others were with the slower dynamo. 
All but No. 4, however, were with a relatively high frequency. 
No. 4 was made using a two-pole rotary transformer, sup- 
plying it with direct current, and running it at a speed of 
1600 per minute. The percentage losses (o) vary more among 
the different condensers at the same frequency than one would 

The percentage loss at the frequency 28 (Experiment 4) is 
'78 per cent., whereas at a frequency five times as great it is 
(Experiment 3) 1*00 per cent. At 120 it is, as would be ex- 
pected, nearly as great as at 140; that is, it is *96 per cent. 


Messrs. Rosa and Smith on a Calorimetric 

Table III. — Measurements of Energ) T 









No. of 

Period in 



in tempe- 

Total heat 



per second 

seconds for 



rature of 




ture of the 

ture of the 




1000 grams 



1000 grams 


of water. 



of water. 





h m s 
7 24 00 





7 59 50 







8 36 30 








9 13 30 







12 33 25 

1 04 00 








1 36 00 




5 39 



2 07 40 







2 40 00 








7 17 20 

7 46 20 








8 15 50 







8 45 23 







9 14 43 


16 55 






12 21 12 


1 21 25 


18 45 





2 37 30 







3 47 35 








12 24 20 


12 55 10 







1 26 45 







1 59 30 








6 05 08 


6 41 30 


17 82 





6 54 30 

780 1 




1739 1 



2 17 55 

2 50 00 




4 39 




3 21 30 







3 53 28 







4 25 00 







* The amount of water for this period was 700 grams. 
t „ .. „ 370 „ 

Determination of Energy Dissipated in Condensers. 235 

dissipated in Stanley Paraffin Condensers. 



= average 
calories x J 

(h) x 4-1972. 



J 3-23 

3 47 










in volts. 

in amperes. 






(i) + (m). 





































Per cent. 

loss = 

■k cot <p X 100. 


Efficiency = 


0)xttx1oo. loo- (o) 










99 00 





236 Prof. K. Pearson on certain Properties 

(Experiment 2). Condenser No. 1, of the same lot, shows a 
loss of 1*45 per cent, at frequency 140, which is 45 per cent, 
greater loss than Nos. 3 and 4 give. Of the second lot, No. 6 
gives a large loss (Experiment 5), and other experiments 
which one of us has made by other methods show that all the 
other condensers of this lot have losses nearly the same as No. 6, 
excepting No. 10, which gives the smallest loss of any, *74 per 
cent, in one case and '78 per cent, in another (Experiments 6 
and 7). Condenser No. 2 shows by other methods the same 
loss as 3 and 4. Hence we have the following singular 
results : — All of the first lot except one have a loss of 10 per 
cent, on high frequencies, and the exceptional condenser has a 
loss of 1*45 per cent. All the condensers of the second lot have 
substantially the same losses, about 1*5 per cent., and the excep- 
tional one is scarcely more than one half as much as the others; 
the exceptional one of the first lot having the same loss as all but 
one of the second. There is no possibility of a confusion of 
numbers, for they were plainly stamped when purchased, and 
the capacities of the first and second lots are very different, as 
already stated. Our experiments do not indicate the reason 
for these large differences ; but the existence of such differ- 
ences is fully confirmed by measurements made by wholly 
independent methods, and which will shortly be published. 

Wesleyan University, 
Middletown, Conn., Sept. 1, 1898. 

XV. On certain Properties of the Hyper geometrical Series, 
and on the fitting of such Series to Observation Polygons in 
the Theory of Chance. By Kabl Pearson, F.R.S., 
University College, London*. 

1. TN a paper entitled " Mathematical Contributions to 
JL the Theory of Evolution : Part II. Skew Variation 
in Homogeneous Material " t, I have pointed out that the 
following series, of which the skew-binomial is a special case 

pn(pn-l) (pn — 2) .... (pn— r + 1) 
n(n — l)(n — 2) (n — r+1) 

( x ! r 9 n , r(r-l) gn(qn-l) 

V pn — r + 1 1.2 (pn — r + l)(pn—r + 2) 

r(r-l)(r-2) qn{qn-l){qn-2) ^ 

1.2.3 {pn — r + l)(pn — r + 2)(pn — r + 3) 

* Communicated bv the Author. 

+ Phil. Trans, vol. clxxxvi. p. 360 (1895), 

of the Hyper geometrical Series. 237 

is especially adapted for fitting various types of frequency- 
distribution. The relative magnitude of r and n is, indeed, 
often a very good test of the " interdependence of contribu- 
tory causes." 
If we put 

a=—r,/3=—qn,y=pn — r+l ... (2) 

and denote by F(a, /3, y, x) the general hypergeometrical 

1+ ^ + ^±l»+i)^ +&c . . . . (3) 

1.7 1.2.7(7+1) 

we see that our series is a hypergeometrical series of the type 
F(a, j3, 7, 1), or, as we shall denote it F 1 (a, ^7), multiplied 
by a factor, which we may write A . 

If the successive terms of a hypergeometrical series be 
plotted up as ordinates at intervals c, and the tops of these 
ordinates be joined, we obtain a great variety of polygons, 
which approximate to the interesting series of generalized 
probability-curves with which I have already dealt. The 
advantage of the hypergeometrical polygons over the curves 
consists in the knowledge as to the nature of the chance 
distribution indicated by the discovery of the actual values 
of p, q, n, and r. The curves, however, possess continuity 
and are easier of calculation. Clearly a knowledge of a, /3, 
y, since 7 — a— /3 — l = n, gives n, and hence g, r, and p. 

We shall find it convenient to write 

m 1 = a + ^, m 2 = a/3. . . . . . (4) 

It is not, however, only in the question of distribution of 
frequency that hypergeometrical series may be of service ; 
it seems extremely probable that the three constants #, /3, 7 
of Fj (a, (3, 7) may be of service in indicating close empirical 
approximations to physical laws, owing to the great variety 
of forms that the hypergeometrical polygon can take. 

Before we proceed to the fitting of hypergeometrical 
polygons to given data, we require to demonstrate one or two 
general propositions with regard to such figures. 

2. On the moments of Fi(a, /3, 7). — Let A= the area 
of the polygon, thus if the ordinates are plotted at distance 
c, we have A = cxF 1 . Let fx s K be the sth. moment of 
the polygon about its centroid-vertical, the elements of area of 
the polygon being concentrated along the ordinates. Let v 8 'k. 
be the sth moment of the ordinates about a vertical parallel 
and at a distance c from the first ordinate, i.e. 

238 Prof. K. Pearson on certain Properties 

Now let a new series of functions xoy %\> Xii & c - De 
formed, so that 

andletx =F(a, /3, y, a). 
Then we have 

v s = c s x(xJXo) x= r 
fi s can then be found from v s v 3 _ v v s _ 2 , &c. by the formulas 
given on p. 77 of my memoir (Phil. Trans, vol. clxxxv.). 

Thus the determination of the successive moments of the 
hypergeometrical series F t is thrown back on the discovery 
of the %'s from the value F. 

3. To find the successive %'s. — The hypergeometrical series 
is known to satisfy the differential equation 

x d / d¥\ . ,„ . x dF ^ . 

(see Forsyth, ' Differential Equations/ p. 185). 
But dF 

Hence, substituting and rearranging, we have 

(l-^){% 2 +(m 1 -2)xi+(m 2 -m 1 + l)%o}+wXi-( n - , -^)Xo = 0. (6) 
Put #=1, we have 

(%i)i=-^- (Xo)i (7) 

or rc + m 2 /ON 

This is the distance of the centroid- vertical of the hyper- 
geometrical series F 1 from the vertical about which the 
v-moments are taken. 

Multiply (6) by x and differentiate, we find 

(l-^){^3 + (m 1 -2)x 2 +(w2-^i + l)%o}-^{X2+( m i-2)x l 

+ (m 2 — m x + l)x<>} + n Xi - ( n + m 2>Xi = 0, 

(1-^){X3+ ( m l~l)%2+(^2-l)%l+(^2~Wl 1 + l)Xo} 

+ ( 7l - 1 )X3-( n + m i + w 2-2)x 1 -(m 2 -w 1 + l)%o = . (9) 

• of the Hyper geometrical Series. 239 

Put a?=I, we have 
( 7l - 1 )(%2) 1 = (n + w 1 + m 2 -2)x 1 + (w 2 ~m 1 4-l)(%o)i 

. . n 2 + n(5m 2 — 1) + m 2 2 + m x m 2 — 2m 2 /1A . 

= (Xo)i , • (10) 

by aid of (7). 

l^ us ri 2 + n(3m 2 — l)+m 2 2 + 77? 1 m 2 --2m2 /1f , N 

v 2 = ^ t fr — . • (11) 

n(n — 1) v 7 

and o 

^ 2 = v 2 -v 1 J 

_^ 2 m 2 (n 2 + m 1 n + m 2 ) 

or, we may write 

c W?(n + «)(n + / 3) (1) 

:* n 2 (n — 1) v y 

Multiplying (9) by # and differentiating again, we find 

{l-x) {x± + m^ +{m 1 + m 2 -2)X 2 +(2m 2 — m 1 )xi + (m 2 —m 1 + l)xo} 
+ (n— 2)^— (71 + 2772! + w 2 -3)x2 — (2w 2 — m 1 )%i~(m 2 —m 1 + l)xo = 0. (14) 
Putting a?=l, we have 
(n-2)(x 3 )i=(n + 2m 1 + m 2 -3)( %2 ) 1 + (27722-77?!) (^Oi 

+ (2722-772! + 1) (% )l, 

or, by aid of (7) and (10), 

v 3 = c 3 {?2 3 + n 2 (7m 2 — 3) + n (6tt2 2 2 + 67Bi772 2 — 15m 2 + 2) + m 2 3 

+ %m Y m 2 2 + 2m l 2 m 2 — 7m 2 2 — Gm^g + 6m 2 }-4-w(?2 — 1) (72 — 2) . (15) 

Hence, since fi z = v 3 — Sv^— Vi 3 , we have after some reduc- 

_ c 3 xj3(n + *)(n + {3Xn + 2a)(n + 2l3) 
P»- » 8 (»-rl)(n-2) ' ' ' U j 

Differentiating (14) after multiplication by a?, we find 

(l-^)|X5+K + l)X4+(2m l + m2-2)x3+(3m 2 -2) %2 

+ (3m 2 — 2m 1 + l)xi+(w 2 — m 1 + l)xo} + (w— 3)^4 

— (71 + 3772! + m 2 — 3)X3~(3w 2 - 2)% 2 —(3772 2 — 2772! + 1)^! 

- (7722-772! + 1) %0 = (17) 

Putting * =3=1 j we have 

(n - 3) ( X4 )x = (n + 37?ii + m, - 3) Gfc) i + (3m 3 - 2) ( %2 ) j 

+ (3m, — 2m 1 + l)(xi)i+ (wia-mx+lXxoJi. 

240 Prof. K. Pearson on certain Properties 

Hence, by the use of (7), (10), and (15), we deduce 

v^ = c 4 {n 4 + n 3 (15m 2 — 6) + rc 2 (25m 2 2 + 25m 1 m 2 — 65m 2 -4- 11) 

+ n(10w? 2 3 +30m 1 ?72 2 2 + 2()m 1 2 ?^ 2 — 75m 2 2 -65m 1 w 2 + 80m 2 --6) 
4- m 2 + 6w 1 m 2 3 + llwi 1 2 ?w 2 2 + 6m l 3 m 2 — 16m 2 3 — 42wiW 2 2 
— 24m 1 2 /?? 2 -f- S6m ] m 2 + 49w 2 2 — 24m 2 } 

-i-n(n-l)(n-2)(n-3). . . (18) 

But 2 

thus we find 

c 4 m 2 (n 2 + m 1 n-f w? 2 ) . , ,,., „ ., 

* = n'(»-l)(n-2)(n-V X {**+«>(*»» + *".*U 

+ rc 2 (3mim 2 + 6mi 2 + Qm 2 ) + n(3m 2 2 + lSm^m^ + 18m 2 2 }. (19) 
Now a = — r, /3 =—qn ; 

m x r r\ Wo . 

n \ ' n> n 

Substitute these values in (19), and make n infinite. The 
hypergeometrical series now becomes the binomial (p + q) r i 
and we have 

/*t = c\l + 3(r-2)pq), 

a result already deduced (Phil. Trans, vol. clxxxvi. p. 317). 
This serves to confirm (19). 

Dividing equation (19) by (1 — #), and putting = 1, we 
find, by remembering that 

r r s)„=(^)„r{S}„"- te " , " : 

(" - 4) to) l => + 4m! + m 2 - 2) ( %4 ) x + (2mj + 4m 2 - 4) ( %3 ) x 
+ (6m 2 -2m!-l)( %2 ) 1+ (4m 2 -3m 1 +2)( %1 ) 1 

+ (/7? 2 -mi + l)(xo)i . . • 


v 5 — —, tyt 577 577 rr { ft 5 + n 4 (3 lm 2 — 1 0) 

n(n— l)(w — 2)(n— 3)(n— 4) l v 

+ n 3 (90m 2 2 + 90m!m 2 — 220m 2 + 35) 

+ n 2 (65m 2 3 + 195m 1 m 2 2 + 130mi 2 m 2 - 485m 2 2 

-420m 1 m 2 + 535m 2 -50) 


t 2 — uitJ//o.l7l 2 ""Oil 

+ 800m 2 2 -490m 2 + 24) + m 2 5 + 10miW 2 4 

-f 35m l 2 m 2 * + 50m l 3 m 2 2 + 24ra 1 4 m 2 — 160m 1 m 2 3 

- 256mi 2 m 2 2 -i20m 1 3 m 2 + 240™^ + 490/*!m 2 2 

-240m!m 2 - 30m 2 4 + 213tw 2 3 - 380m 3 2 + 120m 2 }, (20) 

+ n(15m 2 4 + 90??*im 2 3 +165m 1 2 m 2 2 + 90mi 3 m 2 

645m.m 2 2 — 370m 1 2 -m 2 -f 600m 1 m 2 

of the Hyper geometrical Series. 


Pi = v 6 — 5 Vi/^4 — 10 vfps — 10 j/ x 3 ^2 — V 

_ c 5 m 2 (n* -f ti^ 4- 77? 2 ) (t? 2 + 2m 1 n + 4m 2 ) 
7i 5 (n-l)(>i-2)(n-3)(7i-4) 

X {n 4 + ?i 3 (10?7i 2 +12m 1 + 5) +w 2 (10m 1 m 2 + 127/i l 2 ) 
+ n(10m 2 2 + 24m 1 m 2 ) + 24m 2 2 }. . . 
To determine the value of n, m u and m 2 , let us write 

A=A t 3 2 /^ 3 , A=/*V/*s 8 i ^3=^5/(^3^2) ; 

and to render the elimination easier, let us put 
e = w 2 -f nmx + m 2 , ' 

m 2 e = 2 2 , V (22) 

e-\-m 2 — z x . 
Then, from (13) and (14), 


A-(Mt ( ^-^ 


From (19) and (14), 

From (21), (13), and (13), 

ft = (^p^ {^ + 5 ^+^(10n--24) + 12(* 1 »-n«* l )}. (25) 

These are linear in 2 2 ; collecting all the terms of ^ 2 on the 
left, we can rewrite (23) to (25), 




— ^4 

7l 4 + 4^ 1 2 -7l 2 < 2r 1 ), 


zA/3 2 {n ^n-3) _ (3h _ 6) J = »*+!!■ + e^-^), ( 27 ) 

z 2 {g 3 (n ""^" 4) -(10n-24) 1 = n« + 5n'+12fa»-n« g| ). (28) 

Multiply (26) by 3, and (27) by 2, and subtract the results 
from (28) : 



(n-3)(n-4) „„ (n-2) 2 



!L_2L_(i0n_24)l=-2» 4 + 5n 3 , . (29) 

^k (W - 8) _ ( r 4) - 2 /3^ "-^r 3) -(4n-12)U-n 4 + 3n3. 


242 Prof. K. Pearson on certain Properties 

The last equation will divide by n — 3, or 



Substitute this value of n 3 on the right-hand side of (29), 
and we have 

A ( w -3K,-4) _ 3A (^-2) ! _ (1()n _ 24) 


w 2 

Divide out by and we have 

J n — 1 

&(n-4)-2A(2/i-5) + 3ft(n-2) + 2(n-l)=0, 

^(A-4 y 5 2 + 3/3 1 + 2)=4y5 3 -10A + 6/3 1 + 2. 

4^3-10^ + 6^ + 2 

n - A-4ft + 3A + 2 (32) 

n being now known (31) gives us 

n*(n — 1) ,„„v 

* 2 -4(rc-l) + 2/3 2 (^-2)-/3 3 (rc-4) ; " ' K ° 0) 
z 2 and n being known, we have bv (23) 

^-kW^s^ • • • ^ 

Then ra 2 and e are the roots of the quadratic 

? 2 -2i? + *2=0 (35) 

m 2 and e being known, we have 

e — m 2 — n 2 
™i = " (36) 

Next, « and /3 are roots of 

f2_ mi ^ +m2==0 , (37) 

and y=n + « + /3+l. . ..... (38) 

Lastly, from (2) 

r=-«, q=-fi/n i and ya ?"*" 1 , , (39) 

of the Hypergeometrical Series. 243 

Thus, all the constants p, g, r, and n of the series (1) are 

The base unit c is given by (12), or 

e=w Je&=R (40) 

To obtain the successive ordinates of the hypergeometrical 
frequency-polygon we must, if A be the total number of 
observations, take the successive terms of 

1 + r qn + T(r " 1] - 9<gn-l) &c> 

pn — r+1 1.2 (pn — r+1) [pn— r + 2) 

multiplied by A, or 

Apn(pn — l)(pn — 2) . . . (pn— r + 1) 
c n(n — l)(n — 2) . . . (n — r + 1) 

The position of the first ordinate is at a distance d^v x —c 
from the mean (or centroid- vertical) of the series, i. e. 

d=cm 2 /n (41) 

Thus the solution is fully determined. Its possibility 
depends on positive and real values for n and r, and for p 
and q. 

As an illustration I take the following data provided for 
me several years ago by members of my class on the theory 
of chance at Gresham College. 

In a certain 18,600 trials the distribution of frequency was 

759 cases of occurrence, 







































Taking moments round the point corresponding to three 
occurrences I find 

/V=- -501,4516, /V= 8-433,6021, 

/*,'= 1-815,5376, /*,'= -10-504,6774. 

^/= -1-948,5484 

244 Prof. K. Pearson on certain Properties 

Thus the mean is : 


and transferring moments to this mean, we have 

^2 = 1-564,0839, 
ya 3 = -530,4806, 
^ = 7-903,2620; 

/3 X = -073,5460, 
&= 2-891,9091, 
/3 3 = 9'525,2597. 

Substituting in (32) we find 

n = 65-203,378. 
Hence by (33) 

^ = 451,811-067, 
and (34) 

^ = 2839-1404. 

S 2 - 2839-1404?+ 451,811-067 = 0. 

This leads to 

m 2 = 169-2229, 

6 = 2669-9175. 
Whence by (36) 

mj=— 26-85115. 
Thus (37) is now 

? 2 + 26-85115f+169-2229, 

a= -10-10546, 

/3= -16-74569. 
Then from (40) we find 

and from (41) 


Thus we conclude that the frequency may be represented 
by a hypergeometrical series of which the start is *0244 before 
zero occurrence, the base unit is '9721, and the mean is at 
2-4985. Further, from (39) 

r= 10-1055, 
jt> = -7432, £ = -2568; 


j9tt=48-4577, ^=16-7457. 

Further, we conclude that the range .of frequency cannot be 

of the Hypergeometrical Series, 245 

greater than 10, and the whole distribution might be closely 
represented by drawing 10 balls 18,600 times out of a bag 
containing 17 white and 48 black balls, and counting the 
white occurrences in each draw. Actually the frequency was 
obtained by drawing 10 cards out of an ordinary pack of 52 
and counting the hearts in each draw. Thus we have : — ■ 

Actually. From theory. 

Start at -'0244 

Mean 25 2 4988 

Number drawn .. . 10 10-1055 

Base unit 1 '9721 

p -75 -7432 

q '25 -2568 

n 52 65-2034 

Now it is clear that the first six results are in good agree- 
ment, but that n diverges from its actual value by 25 per cent, 
although the number of trials, 18,600, is far larger than are 
recorded in most practical cases. 

It is of interest to record the actual and theoretical 
frequencies : — 

No. of hearts. Observed. Theory. 

■0 ...... 759 747-5 

1 3277 3239 

2 5607 5642 

3 .. .. 5159 5172 

4 ...... 2701 2743 

5 907 87L 

6 165 166 

7 24 18-5 

8 1 1 

y o o 

10 : o o 

The deviations are four positive and four negative, and 
four above and four below their respective probable errors. 
Thus the experimental results are in good accordance with 

Notwithstanding this, n has a large deviation from its 
theoretical value when determined by moments. It is clearly 
a quantity, when thus determined, liable to very large probable 
error. Thu«, while the problem is theoretically fully solved — 
ana 1 it is difficult to believe that any other solution can have 
less probable error — yet we meet, unless we take an immense 
number of trials, with large variations in our estimate of the 
number from which the drawing is made. I have tested this 

Phil. Mag. S. 5. Vol. 47. No. 285. Feb. 1899. S 

246 Lord Rayleigh 071 James BernouilU's 

on a variety of series in games of chance and on biostatis- 
tical data, — a small change in a high moment makes a large 
change in n. Accordingly we are liable to form quite 
erroneous impressions of the nature of the hypergeometrical 
series, and even to reach impossible values for p, g, and r 1 
which are determined through n. Thus the problem, which 
is practically an important one, as enabling us to test the 
sufficiency of the usual hypothesis, n = c© , of the theory 
of errors, i. e. to test the " independence or interdependence 
of contributory causes," is seen to admit of a solution, but 
one which is hardly likely to be of much service unless in the 
case to which it is applied a very large amount of data is 

XVI. On James Bernoulli? s Theorem in Probabilities. 
By Lord Rayleigh, F.R.S* 

IF p denote the probability of an event, then the probability 
that in jjl trials the event will happen m times and fail n 
times is equal to a certain term in the expansion of (p+qY> 

■A^-.P^", (i) 

where p+q = l, m + n=r/ji. 

" Now it is known from Algebra that if m and n vary 
subject to the condition that m-f n is constant, the greatest 
value of the above term is when m/n is as nearly as possible 
equal to p/q, so that m and n are as nearly as possible equal 
to fip and fxq respectively. We say as nearly as possible, 
because /xp is not necessarily an integer, while m is. We 
may denote Ihe value of m by pp + z, where z is some proper 
fraction, positive or negative ; and then n = pq—z" 

The rth term, counting onwards, in the expansion of 
{p+qY after (1) is 



? , t p m ~ r q n+r - > . . (2) 

r I n -j- r ! v ' 

The approximate value of (2) when m and n are large 
numbers may be obtained with the aid of Stirling's theorem, 

^! =/ ^*-^27r)£L+ JL + ...J., . (3 ) 

* Communicated bv the Author, 

Theorem in Probabilities. 247 

The process is given in detail after Laplace in Todhunter's 
•History of the Theory of Probability/ p. 549, from which 
the above paragraph is quoted. The expression for the rth 
term after the greatest is 


v^ j~ fxrz r(n — ni) r z ?- 3 \ ,-. 

mf\ + »w i ~2mJi bW + 6V J *' ' *' ' 


and that for the rth term before the greatest may be deduced 
by changing the sign of r in (4). 

It is assumed that r 2 does not surpass fi in order of mag- 
nitude, and fractions of the order l//x are neglected. 

There is an important case in which the circumstances are 
simpler than in general. It arises when p = q^ J, and /jl is 
an even number, so that m = /z — J/*. Here z disappears 
ab initio, and (4) reduces to 


representing (2), which now becomes 


An important application of (5) is to the theory of random 
vibrations. If /x- vibrations are combined, each of the same 
phase but of amplitudes which are at random either -f- 1 or — 1, 
(5) represents the probability of ifi> + r of them being positive 
vibrations, and accordingly \y^— r being negative. In this 
case, and in this case only, is the resultant + 2r. Hence if x 
represent the resultant, the chance of #, which is necessarily 
an even integer, is 

2e-* 2 /V 


The next greater resultant is (^' + 2); so that when a is 
great the above expression may be supposed to correspond to 
a range for x equal to 2. If we represent the range by dx, 
the chance of a resultant lying between x and x + dx is given by 

e- x2 ^dx 
s/(?tth) {) 

Another view of this matter, leading to (5) or (7) without 
the aid of Stirling's theorem, or even of formula (1), is given 

* Phil. Mag-, vol. x. p. 75 (1S80). 

2-18 Lord Rayleigh on James Bernoulli? 's 

(somewhat imperfectly) in ' Theory of Sound,' 2nd ed. § 42 a. 
It depends upon a transition from an equation in finite differ- 
ences to the well-known equation for the conduction of heat 
and the use of one of Fourier's solutions of the hitter. Let 
J(fjL,r) denote the chance that the number of events occurring 
(in the special application positive vibrations) is \p. ■% r, so 
that the excess is r. Suppose that each random combination 
of /jl receives two more random contributions — two in order that 
the whole number may remain even, — and inquire into the 
chance of a subsequent excess r, denoted by/(/x + 2, r). The 
excess after the addition can only be r if previously it were 
r — 1, r, or r-\-l. In the first case the excess becomes r by 
the occurrence of both of the two new events, of which the 
chance is J. In the second case the excess remains r in 
consequence of one event happening and the other failing, 
of which the chance is J ; and in the third case the excess 
becomes r in consequence of the failure of both the new 
events, of which the chance is 5. Thus 

fQ* + i,r)=if{p,r-l) + Vfar) + lfOhr + I). . (8) 

According to the present method the limiting form of /is to be 
derived from (8). We know, however, that/ has actually the 
value given in (6), by means of which (8) may be verified. 
Writing (8) in the form 

we see that when /x and r are infinite the left-hand member 
becomes 2df/d^i, and the right-hand member becomes ^d 2 f/dr 2 , 
so that (9) passes into the differential equation 

dfi 8 dv l ••••••♦ \m 

In (9), (10) r is the excess of the actual occurrences over \yu. 
If we take x to represent the difference between the number 
of occurrences and the number of failures, x=2r and (10) 

iL-l c ll . . . (in 

In the application to vibrations f(fi, x) then denotes the 
chance of a resultant + o? from a combination of ja unit 
vibrations which are positive or negative at random. 

In the formation of (10) we have supposed for simplicity 
that the addition to /x. is '2, the lowest possible consistently 
with the total number remaining even. But if we please we 
may suppose the addition to be any even number yJ , The 

Theorem in Probabilities. 249 

analogue of (8) is then 

+ ^f^- ) /(^''-^' + 2)+..-+/(^'' + ^'); 

and when ^ is treated as very great the right-hand member 

/(„,,.) {i+^'+^^ '+...+/+ 1} 

+ +/( /i '-2) 2 + l./ i ' 2 } 

The series which multiplies / is (L + l)% or 2^. The 
second series is equal to /u' . 2^', as may be seen by com- 
parison of coefficients of x l in the equivalent forms 

(e x + e- x ) n =2 n (l + %a: 2 + . . .) n 

= 6»* + 7^" 2 > + ^"""^ *(—*)* + .... 

I- . w 

The value of the left-hand member becomes simultaneously ■ 

so that we arrive at the same differential equation (10) as 

This is the well-known equation for the conduction of heat, 
and the solution developed by Fourier is at once applicable. 
The symbol fi corresponds to time and r to a linear co- 
ordinate. The special condition is that initially — that is when 
fi is relatively small — -f must vanish for all values of r that are 
not small. We take therefore 

/(ft*) = -^*-**, (12) 

which may be verified by differentiation. 

The constant A may be determined by the understanding 
that/(yu, r) dr is to represent the chance of an excess lying 
between r and r + dr, and that accordingly 

f + 7(ft r)dr=l. ..... (13) 

*/ — °0 

e ~ z ~dz= s/tt, we have 

&~V(h> ■-•■■■ (U > 

250 On James Bernoulli? s Theorem in Probabilities. 

and, finally, as the chance that the excess lies between r and 
r + dr, 

\Z&-^ ( is ) 

Another method by which A in (12) might be determined 
would be by comparison with ((3) in the case of ?' = (). In 
this way we find 



I~ 2.4.6, 

by Wallis' t 



2*. tr I & 

= \/fc) 


If, as is natural in the problem of random vibrations, we 
replace r by x, denoting the difference between the number of 
occurrences and the number of failures, we have as the chance 
that x lies between x and x + dx 


identical with (7). 

In the general case when p and q are not limited to the 
values -J-, it is more difficult to exhibit the argument in a 
satisfactory form, because the most probable numbers of 
occurrences and failures are no longer definite, or at any rate 
simple, fractions of ft. But the general idea is substantially 
the same. The excess of occurrences over the most probable 
number is still denoted by r, and its probability by f(fi, r). 
We regard r as continuous, and we then suppose that /j, 
increases by unity. If the event occurs, of which the chance 
is p, the total number of occurrences is increased by unity. 
But since the most probable number of occurrences is increased 
by p, r undergoes only an increase measured by 1 — p or q. 
In like manner if the event fails, r undergoes a decrease 
measured by p. Accordingly 

f(n+%r)=pf{n r r-q) +qffar+.p). . . (17) 

On the right of (17) we expand f{p,r—q),f(/j,, r+p) in 
powers of p and q. Thus 

Notices respecting New Books. 251 

so that the right-hand member is 

O + q)f+ i j^ (p 2 q +pq*), or /+ \ n -^. 

The left-hand member may be represented by f+df/dp,, so 
that ultimately 

df 1 ffif .,.#. 

d^=^d? (1&) 

Accordingly by the same argument as before the chance of 
an excess r lying between r and r -{-dr is given by 

— i ^e-^P^dr (19) 

We have already considered the case of p = q= z i' Another 
particular case of importance arises when p is very small, 
and accordingly q is nearly equal to unity. The whole 
number /j, is supposed to be so large that pp,, or m, repre- 
senting the most probable number of occurrences, is also 
large. The general formula now reduces to 

,<l \ e-* 2 ' 2m dr, (20) 

V (257T/H) 

which gives the probability that the number of occurrences 
shall lie between m + r and m + r+dr. It is a function of m 
and r only. 

The probability of the deviation from m lying between +r 

"■ '/ fl i [ r e-^dr= -|- f T '—**> ' ' ' ( 21 ) 
y/ (Jtirm). Jo vttJo 

where T = r/ N /(2m). This is equal to "84: when t=1*0, or 

r — s/{2m) ; so that the chance is comparatively small of a 

deviation from m exceeding + s /(2m). For example, if m 

is 50, there is a rather strong probability that the actual 

number of occurrences will lie between 40 and 60. 

The formula (20) has a direct application to many kinds of 


XVII. Notices respecting New Books. 
Textbook of Algebra with exercises for Secondary Schools and Colleges. 

By G. E. Fisher, M.A., Ph.D., and I. J. Schwatt, Ph.D. Part I. 

(pp, xiv-f-683: Philadelphia, Fisher & Schwatt, 1898). 
r PHlS is a big book for the comparatively small extent of ground 
**- it covers. The usual elementary parts are discussed up to 
and including simultaneous Quadratic equations, and then, in the 
remaining 80 pages, we have an account of liatio, Proportion, 
Variation, Exponents, aud Progressions. The Binomial Theorem 
for a positive Integral Exponent occupies about a dozen pages, the 
treatment by Combinations being reserved, we presume, for Part 11. 

252 Intelligence and Miscellaneous Articles. 

The text has been very carefully drawn up and should be useful 
for young teachers. The distinction between the signs of operation 
and the signs of quality is very clearly indicated by means of a 
special notation. There is a good chapter on the interpretation 
of the solutions of Problems, such questions as that of the problem 
of the couriers and allied problems being worked out in some 
detail. In some few places one would have expected the Authors 
to have been a little fuller, but the general level is high. The 
exercises are very numerous and well graded. The number is 
intentionally large " that the teacher from year to year may have 
Variety with different classes." There is no mention of Graphs, a 
branch of the subject which Prof. Chrystal's book brings into 
prominence. The book can be resommend^d as a sound treatise 
on the elements of Algebra, and the printers have done their work 
well. The most important typographical error we have come, 
across is on p. 611 line 5, where for " first" read " second." There 
are no answers at the end. 

XVIII. Intelligence and Miscellaneous Articles. 

To the Editors of the Philosophical Magazine. 

IN your September number Dr. Lodge comments on my objection 
to the conclusiveness of the Michelson-Morley aether experi- 
ment, and I should like to point out that his remarks are founded 
on a strange misconception of the nature of my objection. He 
conjectures that I attribute the negative result of the Michelson 
and Morley experiment " to the possible second-order influence of 
a hitherto neglected first-order tilting or shifting of the wave-fronts 
brought about by the undiscovered drift of the aether past the 
earth." But in my communication I pointed out that nay objection 
was not of this nature, but related to the assumption made as to 
the optical sensitiveness of the system of interference-fringes 
relied ou by the experimenters to enable them to measure the 
minute length in question in their experiments. My contention 
was simply that the system of fringes used in the experiment had 
probably a more complex character than was supposed, and that 
therefore its capability of measuring the small length accurately 
was over-rated to an unknown extent. Evidently Dr. Lodge has 
pondered so deeply on aberration problems that in reading my 
paper his thought has got into sone old groove which he has 
unconsciously taken to be the direction of my argument. 

Yours obediently, 

William Sutherland. 

Postscript by Prof. Lodge. — I was not very clear about Mr; 
Sutherland's precise line of argument, nor am I now ; but there 
was an imaginary loophole which others might attempt to get 
through, though Mr. Sutherland, as it now appears, did not; and 
I took the opportunity (not specially opportune as it turns out) of 
indicating that it was closed. 






.U. WM 
MARCH 1899. V. „ r fpV 

XIX. On the Theory of the Conduction of Electricity through 
Gases by Charged Ions. By J. J. Thomson, M.A., F.R.S., 
Cavendish Professor of Experimental Physics, Cambridge*. 

rilHE electrical conductivity possessed by gases under cer- 
JL tain circumstances — as for example when Rontgen or 
uranium rays pass through the gas, or when the gas is in a 
vacuum-tube or in the neighbourhood of a piece of metal 
heated to redness, or near a flame or an arc or spark-discharge, 
or to a piece of metal illuminated by ultra-violet light — can be 
regarded as due to the presence in the gas of charged ions, 
the motions of these ions in the electric field constituting the 

To investigate the distribution of the electric force through 
the gas we have to take into account (1) the production of 
the ions ; this may either take place throughout the gas, or 
else be confined to particular regions ; (2) the recombina- 
tion of the ions, the positively charged ions combining with 
the negatively charged ones to form an electrically neutral 
system ; (3) the movement of the ions under the electric 
forces. We shall suppose in the subsequent investigations 
that the velocity of an ion is proportional to the electric 
intensity acting upon it. The velocity acquired by an ion 
under a given potential gradient has been measured at the 
Cavendish Laboratory by several observers — in the case of 
gases exposed to the Rontgen rays by Rutherford and 
by Zeleny; for gases exposed to uranium radiation or to 

* Communicated by the Author. 
Phil. Mag. S. 5. Vol. 47. No. 286. March 1899. T 

254 Prof. J. J. Thomson on the Theory of Conduction of 

ultra-violet light by Rutherford ; for the ions in flames by 
McClelland and H. A. Wilson; and for the ions in gases 
near to incandescent metals or to the arc discharge by 
McClelland. The velocities in the different cases vary very 
much ; the velocity of an ion in the same gas is much the 
same whether the conductivity is due to Rontgen rays, 
uranium rays, or ultra-violet light; it is much smaller when 
the conductivity is produced by an arc or by incandescent 
metal. Thus the mean velocity of the positive and negative 
ions under a volt per centimetre in air exposed to Rontgen 
rays was found by Rutherford to be about 1'6 cm./sec, while 
for gas drawn from the neighbourhood of an arc-discharge 
in carbonic acid the mean velocity of the positive and negative 
ions was found by McClelland to be only "0035 cm./sec. 

This difference is caused by the ions acting as nuclei about 
which condensation, whether of the gas around them or of 
water-vapour present in the gas, takes place. The power 
of these ions to act as nuclei for the condensation of water- 
vapour is strikingly shown by C. T. R. Wilson's * experi- 
ments on the effects of Rontgen and uranium radiation on 
the formation of clouds, and also by R. v. HelmhohVsf 
experiments on the effects produced by ions on a steam jet. 
If the size of the aopreoation which forms round the ion 
depends on the circumstances under which the ion is liberated 
and the substances by which it is surrounded, the velocity 
which the ion acquires under a given potential gradient will 
also depend on these circumstances, the larger the mass of 
the aggregation the smaller will be this velocity. A remark- 
able result of the determination of the velocities acquired by 
the ions under the electric field is that the velocity acquired by 
the negative ion under a given potential gradient is greater 
than (except in a few exceptional cases when it is equal to) 
the velocity acquired by the positive ion. Greatly as the 
velocities of the ions produced in different ways differ from 
each other, yet they all show this peculiarity. The relative 
velocities of the negative and positive ions differ very much 
in the different cases of conduction through gases ; thus in 
the case of imperfectly dried hydrogen traversed by the 
Rontgen rays, Zeleny found that the speed of the negative 
ions was about 25 per cent, greater than that of the positive, 
while in the case of conduction through hot flames H. A. 
Wilson found that the velocity of the negative ion was 17 or 
18 times that of the positive. In the case of the discharge 
through vacuum-tubes, the measurements which I made of 

* Wilson, Phil. Trans. A, 1897 ; Proceedings of Cambridge Phil. Soc. 
vol. ix. p. 333. 

t R. v. Helmholtz, Wied. Ann. vol. xxvii. p. 509 (1886). 

Electricity through Gases by Charged Ions. 255 

tbe ratio of the charge to the mass for the particles con- 
stituting the cathode rays and those of W. Wien * for the 
ions carrying the positive charge indicate that the ratio of 
the velocity of the negative ion to that of the positive one 
under the same potential gradient would be very large. 
This fact is, I think, sufficient to account for most of the 
differences between the appearances at the positive and nega- 
tive electrodes in a vacuum-tube. Schuster (Proc. Roy. Soc. 
vol. xlvii. p. 526, 1890), from observations on the rates at which 
positively and negatively electrified bodies lost their charges 
in a vacuum-tube, came to the conclusion that the negative 
ions diffused more rapidly than the positive ; other pheno- 
mena connected with the discharge led me later (Phil. Mag. 
vol. xl. p. 511, 1895) independently to the same result. 

We shall now proceed to find equations satisfied by the 
electric intensity in a gas containing charged ions. To simplify 
the analysis we shall suppose that the electric force is every- 
where parallel to the axis of x, and that if X is the value of 
the electric intensity at a point, the velocity of the positive ion 
at that point is &iX, and that of the negative ion in the 
opposite direction & 2 X; we shall suppose that at this point the 
number of positive ions per unit volume is %, the number of 
negative ions n 2 ; let q be the number of positive or negative 
ions produced at this point in unit volume in unit time : the 
number of collisions per unit time between the positive and 
negative ions is proportional to n x ii 2 . We shall suppose that 
in a certain fraction of these collisions recombination between 
the positive and negative ions takes place, so that a number 
un\n 2 of positive and negative ions disappear in unit time 
from unit volume in consequence of the recombination of the 
ions. If e is the charge carried by each ion, the volume 
density of the electrification is (n x — n 2 ) e, hence we have 

_ =A7r(n 1 -^n 2 )e ! , (1) 

if i is the current through unit area of the gas, and if we 
neglect any diffusion except that caused by the electric field, 

^n^X + Ji 2 n 2 eK. = i, (2) 

and if things have settled into a steady state, i is constant 
throughout the gas; from these equations we have 

1 f l , k 2 afX | .... 

" ie= /tT+I 2 lx + i^j' • • • ( 3 ) 

* W. Wien, Verhandl. der phys. GeseUsch. zu Berlin, vol. xvi. p. 165. 


256 Prof. J. J. Thomson on the Theory of Conduction of 

In a steady state the number of positive ions in unit volume 
at a given place remains constant, hence 
-^(k 1 n 1 X)=q—an 1 n 2 , (5) 


— — (k 2 n 2 X) =9. — *n x n 2 . 

Substituting in either of these equations the values of n Y n 2 
previously found we get, since di/d,v = 0, 

lire k^k.dA lx)~ 9 XVft + ifc^V 4tt rfar/V ** dx t 
If we put X 2 = 2y, 

this equation becomes 

1 ^1^2 ^P * i h \/ ___h \ 

Aire^ + k, P dy ~ q 2ye' 2 (k l + k 2 ) 2 \ ±ir P )\ ^ P )' 

I have not been able to integrate this equation in the 

feneral case when q is finite and k x not equal to k 2 . We can, 
owever, integrate it when q is constant and k x =:k 2 = k. In 
this case the equation may be written 


d I k * 2 2 \ « 1/ ^ 2 gV J. 

the solution of which is 

where C is a constant of integration. 

If the current through the gas passes between two parallel 
plates maintained at a constant potential-difference, d~K/dx = 
midway between the plates ; at the positive plate n x — 0, while 
w 2 = at the negative plate; hence if X , X x be respectively 
the values of X midway between the plates and at either plate, 
we have, putting p = 0, to get X 


Electricity through Gases by Charged Ions. 257 


fc=2n&eX , 

since when dX/dx = 0, n 1 =n 2 . 

The measurements of X for gases exposed to Rontgen rays 
show that unless the current is approaching the maximum value 
it can attain, X is practically constant for some distance near 
the middle of the plates; hence in this case we have d 2 X/d% 2 = 
midway between the plates, and therefore by equations (3) 
and (5) q=a?i 2 ; substituting this value of n we have 



-X* — =CXo= (6) 



At either plate ^2 = 0, so that JQ-2P 2 — * 2 = 0, thus 

-X?_J^ =CX 1 i^ ; .... (7) 



hence from (6) and (7) 

.„ Sirek 
or it 

| i =/5 2 "V (8) 

It follows from this equation that X ly /X is greater than 
unity, and that the value of this ratio increases from unity to 
infinity as /3 increases from zero to infinity. We see that /3 
does not involve either q or i. So that, to take a particular 
case, when the gas between two plates is exposed to Rontgen 
rays, the ratio of the electric intensity at the plates to that 
midway between them is independent of the intensity of the 
radiation and of the current through the gas. The curves 

258 Prof. J. J. Thomson on the Theory of Conduction of 

giving the connexion between the electric intensity and the 
distance between the plates are found by experiment to be 
somewhat as represented in fig. 1. The variation in X (fig. 1) 

Fisr. 1. 

occurs only in two layers near the plates and X is approximately 
constant in the rest of the field. As the current through the 
gas increases the layers of inconstant X expand until they 
touch, and then there is no longer a region in which X is 
constant. We can easily find an inferior limit to the value 
of X the thickness of one of these layers when we are given 
the value of the current. For suppose P (fig. 1) is at the 
boundary of the layer next the positive electrode, then at P, 
since X becomes constant, half the current must be carried 
by the positive and half by the negative ions; if i is the 
current and e the charge carried by an ion, then ij'le positive 
ions must cross unit area of a plane through P in unit time, 
so that at least that number must be produced in unit time in 
the region between P and the positive plate. Now if X is 
the thickness of the layer, qX is the number of positive ions 
produced in unit time; the number that cross the plane in 
unit time cannot then be greater than qX, and will only be as 
great as this if no recombination of the ions takes place ; 



2e : 


Thus t/2eq is an inferior limit to X. It will not, however, I 
think be very far from the true value, for we can show that 
but little recombination will take place in the time taken by 
the positive ions to traverse a layer of this thickness. For 
the rate of combination of the positive ions is given by 

dn x 

= — an l ?i 2 . 

Electricity through Gases by Charged Io?is. 259 

If N 2 is the maximum value of n 2 , then -^=- measures the time 

that is taken before recombination diminishes the number of 
the ions to any very appreciable extent. In this time the 
positive ions would move a distance 8 given by the equation 

where X t is the value of X at the plate. 

&x 2 x,<?=^ 



Let X x =7X0 where 7 = /3 2 - 2/ < J , 
^ yWX*e 


r^_7 2 x 

±qe 2 

Thus if 7 is tolerably large, the positive ions will traverse 
a space much greater than A, before recombining. 

The greatest current that can pass between the plates is 
when all the ions are used in carrying the current; if I is the 
distance between the plates, then Iq positive and negative 
ions are produced in unit time ; thus if I is the maximum 
current which can pass between the plates 

hence we can write 

I = lqe; 

\ t 

The equations 


,^=47r(tt 1 -n 2 )^ 

^(k 1 7i l X)=q-an 1 n 2 , 

J x {k 2 n 2 X) = — (q — ocn^), 
(k 1 n 1 + k 2 n 2 )Xe = c } 

260 Prof. J. J. Thomson on the Theory of Conduction of 

are satisfied by 

7i l = n 2 = {q/a}i y 

k 2 n 2 ~K.i 



^l + A-2 ' 

e{k x + kc. 

where c is the current through the gas. In this case the 
amounts of the current carried by the positive and negative 
ions respectively are proportional to the velocities of those 
ions. When, however, the current passes between two 
parallel plates, this solution cannot hold right up to the plates, 
For, consider the condition of things at a point P, between 
the plates AB and CD, of which AB is the positive and CD 
the negative plate. Then across unit area at P 

k, i 

k,+ko e 

positive ions pass in unit time, and these must come from the 
region between P and AB ; this region can, however, not 
furnish more than q\, and only as much as this if there are 
no recombinations ; hence the preceding solution cannot hold 
when the distance from the positive plate is less than 

k x + k 2 qe ' 

similarly it cannot hold nearer the negative plate than the 

k 2 i 
k x + k 2 qe 

We shall assume that the solution given above does hold in the 
parts of the field which are further away from the plates than 
these distances ; and further, that in the layers in which this 
solution does not hold, there is no recombination of the ions. 
Let us now consider the condition of things near the positive 
plate between 

x = and x — , — ^ = X say. 

k x + k 2 qe 

Then, since in this region there is no recombination, our 

Electricity through Gases by Charged Ions. 261 

equations are 

— = 4:7r{n 1 -n 2 )e, 

^{k 2 n 2 X)=-q. 

If q is constant, we have 

A: 1 ?z 1 X = got, 

# 2 ?2 2 X= go:, 

the constant has been determined so as to make ni~0 when 
x=0 ; substituting these values for n 1? n 2 , in the equation 
giving dUL/dx we get 



x --Kf(H,)-f>^ 

the constant may be determined from the condition that 
when x = \ 

^~qe\k 1 + k 2 ) 2J 
from this we find 

~ OC L 2 f , 47T6 £. ._ _ . 1 

Since C is the value of X 2 when w=0, it is the value of X 2 
at the positive plate; if we call Xj the value of X at the positive 
plate and X the value of X between the layers we have 


thus X x is always greater than X , the value of X between the 

If X 2 is the value of X at the negative plate we have 

Thus, if k 2 , the velocity of the negative ion, is very large 

262 Prof. J. J. Thomson on the Theory of Conduction of 

compared with k 1} the velocity of the positive, the value of 
X at the negative plate is large compared with its value at the 

The curve representing the electric intensity between the 
plates is shown in jSg. 2. In this case & 2 /^i is a large 

Fig. 2. 

The fall of potential across the layer whose thickness is X x 
is equal to 

and this is equal to 

1 !Ldx. 





^X 1 X 1 +i^log(v/ / 3+ V1 + /3): 

Hence, if fi is large, the fall of potential across the layer 
whose thickness is \j next the positive plate, is approximately 

Similarly, if 

and if /Si is large the fall of potential across the layer whose 
thickness is X 2 is approximately 

Electricity through Gases by Charged Ions. 263 

The distance between the plates in which the electric 
intensity is constant and equal to X is I— {\ t +\), where / is 
the distance between the plates. Since \ l + \ 2 = i/qe tne ^ a ^ 
of potential in this distance is equal to 

hence if V is the potential-difference between the plates 


X = ^V l 

^° \q) efa + ki)' 

so that 

\q/ cft + y I v k Y + k 2 qe JV k x -+k 2 qe qe J 

This gives the relation between the current and the poten- 
tial-difference between the plates. It is of the form 

y = A* 2 +B t . 

In a paper by Mr. Eutherford and myself in the Phil. 
Mag. for Oct. 1896, a relation between V and i was given on 
the assumption that the electric intensity was constant between 
the plates ; in this investigation I have tried to allow for the 
variation in the electric intensity. The above investigation 
ceases to be an approximation to the truth when the two 
layers touch each other. In this case the current has its 
limiting value Iqe, and there is no loss of ions by recom- 
bination ; we may therefore neglect the recombination and 
proceed as follows. 

Equations (5) become in this case 

^ (k 1 n 1 X) = q, 

^(k 2 n 2 X)=-q. 

If q is constant, the solutions of these equations are 

Ar 1 w 1 X = g-a?, (9) 

k 2 n 2 X=zq{l-x), (10) 

where x is the distance measured from the positive plate and 
I the distance between the plates, for these solutions satisfy 

264: Prof. J. J. Thomson on the Theory of Conduction of 

the differential equations and the boundary conditions »!=<) 
when # = 0, and ?i 2 = when x—l. From the equation 

— =±7r(? h -n 2 )e, 
we have 


where C is the constant of integration. When X has its 
minimum value we see from equation (1) that n^^ — ^; hence 
from (9) and (10) at such a point we have 

k x x 

irtt ( 12 > 

hence if we determine the point Q where X is a minimum, this 
equation will give us the ratio of the velocities of the positive 
and negative ions. 

We see that a positive ion starting from the positive 
plate, and a negative ion starting from the negative plate, 
reach this point simultaneously. 

If X is the minimum value of X, and f the distance of a 
point between the plates from Q, we may write equation (11) 
in the form 

if I is the maximum current, this may be written 

x-x^+w^I+i-). . . . {13) 

We see from this equation that if we measure the values of 
X at two points and I, the maximum current, we can deduce 
the value of 

and since from (12) we know the value of k v >k 2 , we can 
deduce the values of k ± and k 2 . 

If the positive ion moves more slowly under a given 
potential gradient than the negative ion, then we see *from 
(12) that Q is nearer to the positive than to the negative 

Electricity through Gases by Charged Ions. 265 

plate. Hence it follows from (13) that tho electric force at 
the negative plate is greater than that at the positive. 

A very convenient method of determining the velocities of 
the ions, and one which can be employed in nearly every case 
of conduction through gases, is to produce the ions in one 
region and measure the electric intensity at two points in a 
region where there is no production of ions, but to which ions 
of one sign only can penetrate under the action of the 
electric field. Thus let A, B represent two parallel plates 
immersed in a gas, and let us suppose that in the layer 
between A and the plane LM we produce a supply of ions, 
whether by Rontgen rays, incandescent metals, ultra-violet 
light, or by other means, and suppose that the gas between 
LM and B is screened off from the action of the ionizer. 
Then if A and B are connected to the poles of a battery a 
current will pass through the gas, and this current in the 
region between LM and B will be carried by ions of one 
sign. These will be positive if A is the positive pole, nega- 
tive if it is the negative pole. Let us find the distribution 
of electric intensity in the region between LM and B. Let 
us suppose A is the positive plate, then all the ions in this 
region are positive and we have, using the same notation as 

dX A 

k^Xe = t, 

where i is the current through unit area ; from these equations 
we have 

^ dX _ 4ttc 

dx ~~ k l ' 



Hence if we measure the values of X at two points in 
the region between LM and B, and also the value of t, 
we can, from this equation, deduce the value of k u and 
hence the velocity of the positive ion in a known electric 
field. To determine the velocity of the negative ion we 
have only to perform a similar experiment with the plate A 

When the ionization is confined to a layer CD between the 
plates A and B, the distribution of electric intensity is repre- 

266 Prof. J. J. Thomson on the Theory of Conduction of 

sented by fig. 3, where A is the positive and B the negative 
plate, and the velocity of the negative ion is supposed to be 
much greater than that of the positive. 

Ksr. 3. 

The investigation of the distribution of electric intensity 
given on p. 262 shows that when the velocity of the negative 
ion is much greater than that of the positive, the distribution 
of the intensity has many features in common with that 
associated with the passage of electricity through a vacuum- 
tube, especially the great increase in electric intensity close 
to the negative electrode. Thus this feature of the discharge 
through vacuum-tubes can be explained by the greater 
velocity of the negative ion than of the positive, a property 
which seems to hold in all cases of discharge of electricity 
through gases. And as the most important of the differences 
between the phenomena at the two poles of a vacuum-tube 
are direct consequences of the electric intensity at the cathode 
far exceeding that at the anode, I think the most striking- 
features of the discharge through vacuum-tubes are conse- 
quences of the difference in velocity between the positive and 
negative ions. In the case discussed on p. 261 we assumed 
q constant, i. e. that the ionization along the path of the 
discharge was constant ; in the case of the discharge through 
vacuum-tubes, where the ionization is due primarily to the 
electric field itself, it is unlikely that the ionization will be 
constant when the field is so variable. We can derive 
information as to the distribution of the ionization by a study 
of the very valuable curves giving the distribution of electric 
intensity in a vacuum-tube which we owe to the researches 
of Graham (Wied. Ann. lxiv. p. 49, 1898). 

From the equations 

_= 4:7r(n l -n 2 )e, 

Electricity through Gases by Charged Ions. 207 



(k^i^X) =q — an 1 n 2) 

- X. (^ 2 X) =^-«??in 2 , 

we get if k 1 and &o are independent of x 
cf~X 2 


S7re{q-a ni n 2 )(^ + j-J. 

Thus q — a.n Y n. 2 is of the same sign as d 2 X. 2 /dx 2 . Tims when 
^ — a/^/^2 is positive, that is when the ionization exceeds the 
recombination, the curve for X 2 will be convex to the axis 
of x, and this curve will be concave to the axis of x when the 
recombination exceeds the ionization. Places of sharp cur- 
vature will be regions either of great ionization or recom- 
bination. Fig. 4 is a curve for X 2 calculated from Graham's 

Fig. 4. 


results. It will be seen that there arc two places of specially 
sharp curvature with the curvature in the direction denoting 
ionization, one, the most powerful one, just outside the 
negative dark space, the other near the anode, while in the 
positive light the curvature indicates recombination. It 
would seem as if the positive ions formed at the centre of 

268 Conduction of Electricity through Gases by Charged Ions. 

ionization near the anode, in travelling towards the cathode, 
met with the negative ions coming from the centre of 
ionization near the cathode, that these positive ions combine 
with the negative nntil their number is exhausted, and on 
combining give out light, the region of recombination con- 
stituting the positive light. In the dark space between the 
positive light and the negative glow these positive ions from 
the centre of ionization near the anode are exhausted, so that 
there are none of them left for the negative ions coming from 
the centre near the cathode to combine with. 

The nick in the curve denoting the centre of ionization 
near the cathode is present in all the curves given by 
Graham ; the centre near the anode is not nearly so per- 
sistent. In several of the curves given by Graham there is 
no nick near the anode, though the one near the cathode is 
well marked, and in these tubes there is no well-developed 
positive light. The distribution of potential which accom- 
panies the luminous discharge requires a definite distribution 
of electrification in the tube, this requires ionization and a 
movement of the ions in the tube before the luminous dis- 
charge takes place. There must, therefore, be a kind of 
quasi-discharge to prepare the way for the luminous one. 
Warburg ( Wied. Ann. lxii. p. 385) has, in some cases, detected 
a dark discharge before the luminous one passes. It seems 
probable that such a discharge is not limited to the cases in 
which it has already been detected, but is an invariable 
preliminary to the luminous discharge. 

Besides the " nicks " or places of specially sharp curvature 
fig. 4 shows that there is a small curvature in the direction 
indicating an excess of ionization over recombination all 
through the considerable space that intervenes between the 
positive light and the negative glow ; as this region is one far 
away from places of great electric intensity it seems probable 
that in producing ionization the electric intensity at any 
point is helped by other agencies. The case of the cathode 
rays shows that the motion of charged ions tends to ionize 
the surrounding gas. E. Wiedemann, too, has shown that 
the discharge generates a peculiar radiation, called by him 
a Entladungstrahlen"; it is possible that these may possess 
the power of ionizing a gas through which they pass. 

[ 269 ] 

XX. Cathode, Lenard, and Rontgen Rays. 
By William Sutherland *. 

^PO explain the results of his experiments on cathode rays, 
*- and to account for the Hertz-Lenard apparent passage 
of cathode rays through solid bodies according to Lenard's 
wonderfully simple law, J. J. Thomson (Phil. Mag. [5] . 
xliv., Oct. 1897) proposes the hypothesis, that the matter in 
the cathode stream consists of atoms resolved into particles 
of that primitive substance out of which atoms have been 
supposed to be composed. Before a theory of such 
momentous importance should be entertained, it is necessary 
to examine whether the facts to be explained by it are not 
better accounted for by the logical development of established 
or widely accepted principles of electrical science. 

The chief facts which Thomson arrives at from his experi- 
ments are : — That the cathode rays travel at the same speed 
in different gases such as hydrogen, air, and carbonic dioxide; 
and that m/e, the ratio of the mass of the particles to their 
charge, is the same for the cathode streams in all gases, and 
is about 10 -3 of the ratio of the mass of the hydrogen atom 
to its charge in ordinary electrolysis. These seeming facts 
have also been brought out with great distinctness in the 
experiments of Kaufmann (Wied. Ann. lxi. and lxii.). 
Whatever proves to be the right theory of the nature of the 
cathode rays, the quantitative results which these experi- 
menters have obtained (as did also Lenard), in a region, 
where, amid a bewildering wealth of qualitative work, the 
quantitative appeared as if unattainable, must constitute a 
firm stretch of the roadway to the truth. 

Let us briefly consider the theories used by J. J. Thomson 
and by Kaufmann to interpret their experiments. For instance, 
Thomson considers N particles projected from the cathode, 
each of mass ra, to strike a thermopile, to which they give up 
their kinetic energy |Nmv 2 measured as W. Each of the 
particles carries its charge of electricity e, the whole quantity 
Ne being measured as Q. Thus we have 

±v 2 m/e = W/Q (1) 

But again, the particles, after being projected through a slit 
in the anode with velocity v, are subjected to a field H of 
magnetic force at right angles to the direction of motion, so 
that the actual force tending to deflect each particle is Hev 
at right angles to H and v. The result is that each particle 
describes a circular path of radius p with the centrifugal 

* Communicated by the Author. 
Phil. Mag. S. 5. Vol. 47. No. 286. March 1899. U 

270 Mr. W. Sutherland on 

force mv 2 /p equal to H^v, and therefore our second equation 

vm/e=*Rp (2) 

By measuring W/Q and Hp, Thomson is able to deduce 
values of v and m/e from (1) and (2), and these are the values 
which lead to his remarkable conclusions already given. 

Thomson and Kaufmann control the results of this method 
by a second method of experimenting, in which deflexion of 
cathode rays was produced by electrostatic force, as well as 
by magnetic, the forces in Thomson's experiments being 
adjusted so that the deflexions in both cases were the same, 
and therefore, if F is the electric force, 

Fe = Rev (3) 

Thus an independent measure of v is taken, and as it confirms 
those made by the other method, the experimental evidence 
for the simplicity of the laws of cathode motion is greatly 
strengthened. But in the theory of these experiments there 
is one suppressed premiss, namely, that a charge e must be 
associated with some mass m. Now in following up the 
ionic hypothesis as far as it w T ill go, it is our duty to use this 
premiss as one of the links in the chain of reasoning ; but 
when it leads us to a conclusion subversive of the ionic 
hypothesis, namely, that atoms are split up into particles 
having different charges from the atomic charge in electro- 
lysis, then we are no longer bound by the ionic hypothesis. 
It may therefore be that free electrons can appear in the 
sether, and that in the cathode stream the greater part of the 
electricity travels as free electrons. 

A systematic statement of the reasons for contemplating 
the possibility of the motion of free electrons through the 
gether will be given below ; but in the present connexion it 
is of most importance to consider whether such electrons could 
give up to the thermopile the kinetic energy measured by 
Thomson. From the writings of Thomson, Heaviside, Searle, 
and Morton (Phil. Mag. [5] xi., xxvii., xxviii., xli., xliv.) 
we can form an idea as to what takes place when an electron 
is set in motion. These writings relate to electric charges 
.on conducting spheres and ellipsoids, the charges being 
caused to move by the motion of the conductors; but in the 
case of the free electron we cannot say that its charge is on 
anything, unless a modified portion of the sether. Our 
simplest plan is to regard the electron as a spherical shell of 
electricity of total amount e, the radius being a. 

The main effect of setting such an electron in motion by 
means of some source of energy, is that electric and magnetic 

Cathode, Lenard, and Rout gen Rays. 271 

energy are spread into the aether with the velocity of light 
V, so that when the electron has velocity u the total amount 
of such electric and magnetic energy is (Searle, Phil. Mag. 

e 2 /V. Y + u n \ 

If u is small compared with V this is 

and taking account only of the part of this energy due to 
motion, we have Heaviside's result : — - 

*V/3KaV 2 or fie 2 u* /3a. 

Now if the process, by which some of our store of energy 
was converted into electric and magnetic forms on setting 
the electron in motion, is a reversible one, then on stopping 
the electron in a suitable manner the electric and magnetic 
energy ought to flow back to our source or to the stopping 
body, and if there are no arrangements at the stopping body 
suitable for storing this as ordinary kinetic or potential 
energy, it will appear as heat amongst the particles which 
take part in the stoppage. Thus, then, certain actions of a 
moving electron take place as if it had a localized inertia, 
just as in the theory of electric currents a large part of their 
behaviour is such as it would be if the moving electricity 
had localized inertia. According to Searle's expression, the 
inertia or effective mass of the electron becomes a function of 
its velocity, if we define it as the quantity which is to be multi- 
plied by half the square of the velocity to give the kinetic 
energy. With Heaviside's expression for smaller velocities, 
we should have the inertia equal to 2[ie 2 /3a. But apart from 
these details, we have only to assume that the energy 
imparted to an electron when it is set in motion (or the 
greater part of it) is given up as heat to the material particles 
which arrest its motion, and is equal to half the square of 
the velocity multiplied by a certain quantity characteristic 
of the electron and appearing by the symbol m in the equations 
of Thomson and Kaufinann. Then the experimental results are 
at once explained ; for as the negative electrons are the same 
in all the experiments, m/e has the same value for cathode 
streams in all gases : the gas facilitates the electric discharge, 
but does not control it ; as a steam-engine can give the 
same results with several lubricants, so the cathode stream 
can give the same stream of electric energy by means of its 

U 2 

272 Mr. W. Sutherland on 

moving free electrons, whatever may be the gas used to 
facilitate its flowing. 

We can use Thomson's and Kaufmann's value of mje, 
namely, about 10~ 7 when e is measured in electromagnetic 
units, to calculate the order of magnitude of a the radius of 
the electron. With the relation m/e = 2/jie/3a and /jl = 1 and 
e=10~ 21 in the electromagnetic system of units, we then 
have a = 10~ 14 nearly, while the radii of molecules are of the 
order 10~ 8 cm., so that the linear dimensions of an electron 
are about the millionth part of those of molecules. We 
must therefore concede to the electron great freedom of 
motion in the interstices between the molecules even of solid 

A very remarkable fact about the equations of motion of 
the cathode stream used by Thomson is that, although the 
velocity attained is about one-third that of light, there is no 
sign of any necessity to take account of appreciable frictional 
resistance. The electrons stream through the aether with 
nearly the velocity of light and yet provoke no noticeable 
resistance. What wonder, then, that any sethereal resistance 
to planetary motion has remained beyond our ken ! 

The importance of the quantitative results in these experi- 
ments has necessitated their being discussed out of their 
historical and logical order in a train of thought on cathode 
and allied rays, which order we will now attempt to follow 

Stoney's interpretation of Faraday's law of Electrolysis to 
mean that electricity exists in separate natural units, the 
electrons, as definitely as matter in atoms, is now generally 
accepted, after Helmholtz's independent advocacy of it in 
his Faraday lecture. 

Many workers have investigated the general dynamics of 
electrons, but mostly on the supposition that the electron 
must be associated with an atom, so that they form in con- 
junction an ion. But if electric action in matter is to be 
explained only by the participation of electrons, it naturally 
follows that we should contemplate the existence of electrons 
in the sether to enable it to play its part in electrical action. 

And next we have to take account of the hypothesis 
advanced by Helmholtz in his Faraday lecture (Chem. Soc. 
Trans, xxxix. 1881) to explain Contact Electromotive Force, 
namely, that different atoms attract electrons with different 
amounts of force. This hypothesis may not be generally 
accepted yet, but we propose to follow out its logical con- 
sequences. If two things attract one another they must be 
entities of somewdiat the same sort, and therefore the electron is 

Cathode j Lenard, and Rontyen Rays. 273 

of essentially the same nature as an atom. But further, if two 
things attract one another, we must conceive the possibility 
of their being drawn apart, so that the ion can be split into 
an uncharged atom and an electron free of attachment to 
matter. Maxwell's ascription of inertia to electricity, in 
his theory of induced currents, bears out our conclusion that 
the atom and the electron are things of the same sort in many 
respects. If the electrons are distributed through the aether, 
we must suppose that in aether showing no electric charge 
each negative electron is united with a positive electron to 
form the analogue of a material molecule, which might con- 
veniently be called a neutron. Of the existence of neutrons 
in the aether we have powerful evidence in Trowbridge's 
wonderful experiments ( u The Electrical Conductivity of the 
.Ether," Phil. Mag. [5] xliii., May 1897). He opens his 
account of them with a mention of Edlund's old contention 
that the aether is a conductor and J. J. Thomson's refutation 
of it, and closes it with the statement, " My experiments lead 
me to conclude that under very high electrical stress the 
aether breaks down and becomes a good conductor." Thus both 
Edlund's contention and J. J. Thomson's are happily recon- 
ciled ; the aether is a perfect insulator until it is broken down, 
after which it is a conductor. According to the present 
theory, Trowbridge's result would be worded thus : — The 
aether insulates until the electric force at some point is 
sufficient to decompose the neutrons into electrons, where- 
upon it becomes a conductor of the same type as electrolytes. 
This principle should help practical electricians to construct 
a consistent theory of the hitherto rather intractable electric 

But to return to the cathode rays. The volume of experi- 
mental and theoretical work on the ionization of gases, which 
has been turned out from the Cavendish Laboratory, leaves no 
doubt as to the existence of ions in rare gases through which 
a current of electricity is passing : hence in the cathode stream 
there must be a certain number of ions flying along side by 
side with the electrons; but the experiments of Thomson and 
Kaufmann, according to our interpretation, prove that the 
stream of ions is of quite subsidiary importance to the stream 
of electrons. This is not always necessarily the case in the 
electric discharge through gases, and it seems to me that, for 
a satisfactory theory of the varied phenomena of electric 
conduction through gases, we must take account of the fact 
that we have two conducting media participating in the action 
namely, varying numbers of ions and also of free electrons. 

Our theory of the cathode stream has the advantage that it 

274 Mr. W. Sutherland on 

leads in a most natural manner to a theory of the Lenard rays. 
The cathode stream of electrons, moving with a velocity 
nearly that of light, possessing inertia, and yet of a size that 
is small compared to the molecular interspaces in solids, must 
be able to penetrate a solid that is thin enough, and to emerge 
on the other side, differing from the original cathode stream 
only in that the small trace of moving ions has been filtered 
out. Practically then Lenard rays are cathode rays. This 
is what experiment has abundantly proved. All the main 
properties of the cathode rays have been re-observed in the 
Lenard rays: thus Perrin proves that the cathode stream 
carries negative electricity, McClelland proves the same for 
the Lenard rays : Eontgen discovers that where the cathode 
stream strikes a solid it emits Rontgen rays ; Des Coudres 
proves that where the Lenard rays strike a solid they also 
emit Rontgen rays : Goldstein discovered that the cathode 
stream colours salts, especially haloid salts of the alkalis, in a 
remarkable way; Des Coudres proves the same for the 
Lenard stream: and so on with such properties as magnetic and 
electric deflectability, power of exciting luminescence, and 
the like. The cathode and Lenard streams are the simplest 
forms of electric current known to us. Such a power as that 
of causing certain substances to emit light is only another form 
of our fundamental principle, that an electron in having its 
motion arrested imparts energy to the arresting molecules, 
and of course to their associated electrons. The colouring of 
salts discovered by Goldstein would be accounted for by the 
supposition that some of the negative electrons attach them- 
selves to the electronegative atoms, thereby converting them 
into free ions, and liberating uncharged atoms of the metal, 
which cause the coloration. The experiments which have 
been made, with negative results, to detect the metal or the 
ion chemically do not decide anything, because of course the 
amounts produced are too small for ordinary methods of 
analysis to detect. The fatigue, which some substances show 
after fluorescing for a while under the influence of the cathode 
stream, may be' accounted for in a similar manner by the 
lodgement of free electrons, which produce an opposing 
electromotive force and diminish the intensity of the cathode 
stream, while at the same time producing an analogous change 
to the change of colour in the salts studied by Goldstein, 
except that the change does not appear as visible colour, but 
as a lowering of fluorescent power. Fluorescence is known 
to be very sensitive to the presence of small traces of sub- 

We do not know enough of the relations of atoms and 

Cathode, Lenard, and Rontgen Rays. 275 

electrons to formulate a priori what ought to be the law of 
the resistance of bodies to the passage of a stream of electrons 
through them ; but fortunately we have the comprehensive 
investigations of Lenard on the subject and can give a 
reasonable explanation of his results. He found (Wied. Ann. 
lvi.) that for a great variety of substances of densities varying 
from that of hydrogen at 3 mm. of mercury pressure ('0 6 368) 
to that of gold (19*3), the resistance to the passage of Lenard 
rays depended almost solely on density, the coefficient of 
absorption being proportional to the density. Now we should 
expect our electron being so small compared to atoms, and 
moving with high velocities, to deform locally any atom 
which it strikes, and to rebound before the deformation had 
travelled far into the substance of the atom, so that after the 
electron had departed the atom would be left with an increase 
of vibrational energy, but no direct appreciable increase of 
translatory energy ; then, if the velocity of propagation of a 
disturbance in all atoms is the same, and also the time of an 
encounter between atom and electron constant, the energy 
given up by an electron in an encounter with an atom will 
be proportional to the density of the substance of the atom. 
Now in the case of a solid, as an electron threads its way 
through the molecular interspaces, the number of its encounters 
will be proportional to the length of path, and therefore to 
the thickness of the solid, and therefore the coefficient of 
absorption, which will relate to unit thickness of all substances, 
will be proportional to the density of the substance of the 
atom, which is nearly the same as the density of the sub- 
stance ; thus for solids we interpret Lenard's law of the 
absorption of cathode rays. 

In the case of gases an interesting difference presents 
itself. The electron is not now threading its way through 
narrow passages, but has far more clear space than obstacle 
ahead of it. As the electron is very small itself, we may say 
that in passing through a gas the number of times it en- 
counters a molecule is proportional to the mean sectional area, 
and therefore to the square of the radius R of the molecule 
regarded as a sphere, and also to the number of molecules 
per unit volume (n) ; and if m is the mass of the molecule the 
density of its substance is proportional to ?n/R 3 , and thus the 
coefficient of absorption for a gas is proportional to nWrn/W 
or rnn/R; but nm is the density p, so that the coefficient of 
absorption of a gas is proportional to the density, but also 
inversely proportional to the molecular radius. Now this 
theoretical conclusion corresponds partly with one of Lenard's 
experimental results, namely, that although the coefficient of 

276 Mr. W. Sutherland on 

absorption for a large number of gases appeared to be pro- 
portional to the density within the limits of experimental 
error, the coefficient for hydrogen was exceptional to an 
extent decidedly beyond possible experimental error. In his 
experiments, Lenard showed that if J is the intensity of a 
Lenard stream at its source, J that at a distance r from the 
source in a substance whose coefficient of absorption is A, 

J=J «- A 7^, 

and determined A for various gases at a pressure of one atmo. 
As the densities of these gases are as their molecular weights, 
with that of hydrogen = 2, he shows the relation of A to the 
density of different gases by tabulating values of A/m ; while 
according to our reasoning RA/m would be expected to 
be constant. The following table contains Lenard's values 
of 10 3 A/m, and relative values of R as given in my paper on 
the " Attraction of Unlike Molecules — The Diffusion of Gases," 
Phil. Mag. [5] xxxviii., being half the cube-root of the 
limiting space occupied by a gramme-molecule of the sub- 
stance and controlled by comparison with molecular dimen- 
sions as given by experiments on the viscosity of gases ; the 
last row contains the product 10 3 RA/m : — 

H 2 . CH 4 . CO. C 2 H 4 . N 2 . 2 . C0 2 . N 2 0. SO a . 

10 3 A/m ... 237 124 122 132 113 126 115 102 133 

R , 1025 147 1-35 1-75 1415 134 156 1-535 1-63 

10 3 RA/w... 243 182 165 231 160 169 179 157 217 

Thus while Lenard's approximate constant ranges from 102 
to 237, the one to which we have been led ranges from 157 to 
243, which is an improvement. The really striking point 
about Lenard's discovery, however, is that when A is divided 
by density, the range in value is from 2070 for paper to 5610 
for hydrogen at one atmo ; the results for many substances 
such as gold and hydrogen at 1/228 atmo falling between 
these extremes. The fact that the value of Afp for a rare 
gas is almost the same as for a dense solid, would seem to 
indicate that it is only when an electron strikes an atom 
almost in the direction of a normal that the most important 
part of the absorption of energy occurs; for if this is so, 
the chance of an electron's encountering an atom in a solid 
normally, while threading its way through the interstices, 
being the same as if it could pass through all the atoms which 
it does not meet normally, the absorption of energy from an 
electron by a number of atoms should be the same whether 
they are as close as in a solid or as wide apart as in a rarefied 
gas. Thus probably the coefficient of absorption for a solid 

Cathode^ Lenard, and Rbntgen Rays. 217 

depends on its molecular radius, but the data hardly permit 
of an examination into this point. It should be remarked 
that, as the electron can probably pass easily between the 
atoms in a molecule, the absorption due to a compound mole- 
cule ought to be analysed into the parts due to its atoms ; 
for instance, in Lenard's table of values for A in such gases 
as CH 4 , C0 2 , C 2 H 4 , the part due to the carbon, the hydrogen, 
and the oxygen, ought to be separated out, and then each 
part ought to be proportional to the atomic mass and inversely 
proportional to the atomic radius. If this is so, then the 
agreement in the values of RA/m in our table could not be 
expected to be perfect. 

A very characteristic property of the Lennrd rays follows 
from our theory, for when the cathode rays fall on an alu- 
minium window, such as Lenard used, they have a direction 
normal to it, whereas the Lenard rays issuing on the other 
side of the window are uniformly radiated in all directions ; 
and this is exactly how our stream of small electrons would 
behave, because after they have threaded their way through 
the molecular interstices, they will issue with directions 
uniformly distributed in space, for it is to be presumed that 
the final directions of the intermolecular passages will be 
distributed at random. 

As the cathode and Lenard streams are currents of electrons, 
and therefore form pure electric currents, we might expect 
a priori that the coefficient of absorption of substances for 
them would show some decided relation to the electrical 
resistance of the substances ; but Lenard's law proves that 
such an expectation would be futile, for the absorption of 
conductor and insulator alike depends almost entirely on 
density. This fact throws considerable light on the nature 
of metallic conduction. It would seem as if in the conduction 
of electricity in metals, both the positive and negative elec- 
trons, distributed through the metals, take part in the process 
of conduction, probably in the method of the Grothuss chain; 
by a process of exchange of partners both kinds of electron 
get passed along in opposite directions without anything of 
the nature of a great rush of one kind of electron at one time 
and place. When such a rush occurs in a cathode stream, 
the internal appliances of the best conducting metal can no 
more facilitate its passage, than can the obstructing appliances 
of the best insulator hinder it. In metallic conduction we 
have to do with a property of the metallic atom, whereby, 
with the aid of electromotive force, the local dissociation of 
the neutron into electrons is greatly facilitated ; whereas in 
insulators the reverse is the case. This important field of the 

278 Mr. W. Sutherland on 

relations of electrons and atoms must be nearly ready for 
important developments. 

Two more of Lenard's facts are of special importance, 
namely, that cathode rays, when passed through a window 
from the vacuum-tube in which they are generated, travel as 
Lenard rays through gas of such a density as would prevent 
the formation of the cathode rays, if it prevailed in the tube, 
whether that density is great, as in the ordinary atmosphere, 
or very small, as in a vacuum so high as to insulate under the 
electric forces in the tube. These facts are explained by our 
theory : the properly exhausted tube furnishes a requisite 
facility for splitting up the neutrons and getting a supply of 
electrons to be set swiftly in motion ; once that is accomplished, 
nothing will stop them until it offers enough resistance to 
destroy the momentum of the electrons, and ordinary lengths 
of dense or rare air in Lenard's experiments failed to do this. 
The action of the tube in generating the cathode rays may be 
likened in this connexion to a Gifford's Steam Injector. 

In the logical development of the present line of thought, 
an attempt at an explanation of the cause of the Rontgen 
rays must find a place. Already J. J. Thomson, in his paper 
on a Connexion between Cathode and Rontgen Rays (Phil. 
Mag. [5] xlv., Feb. 1898), has worked out in some detail the 
electromagnetic effect of suddenly stopping ions moving with 
high velocity, the main result being that thin electromagnetic 
pulses radiate from the ion. He believes that these pulses 
constitute the Rontgen rays, in agreement with a surmise of 
Stokes. Thomson's reasoning would apply to our free 
electrons just as to his ions, but there would be this important 
distinction, that while Thomson's hypothesis involves the 
condition that the greater part of the energy of the cathode 
stream consists of the kinetic energy of the atoms, in our 
hypothesis the energy belongs almost entirely to the moving 
electrons, and when these are stopped the energy appears as 
heat at the place of stoppage. Thus Thomson's electromag- 
netic pulses appear only as subsidiary phenomena in con- 
nexion with the conversion of the kinetic energy of the 
electrons into heat ; indeed, we cannot be sure that they 
exist, because their existence has been suggested only in 
accordance with the particular assumptions in Thomson's 
hypothesis which correspond to only a limited portion of the 
complete electrodynamics of such an action as is contemplated 
in this paper, causing the conversion of all or almost all the 
kinetic energy of an electron into heat. Moreover, in tracing 
the relation of Lenard rays to cathode rays we have been led 
to picture the stoppage of the moving electrons as nothing 

Cathode , Lenard, and Rontgen Rays. 279 

like so sudden as that which Thomson has to contemplate for 
his charges : this of course only makes a difference in the 
degree of intensity of the phenomena resulting from the 
stoppage. There has as yet been no systematic proof that 
the properties of a train of impulses would be the same as 
those of the Rontgen rays in the matter of the absence of 
refraction and reflexion. Again, it is recognized that the 
Rontgen rays and the Becquerel rays from uranium are very 
similar, but it would be hard to imagine the Becquerel rays 
to be due to thin impulses. On these grounds it seems to 
me that Thomson's suggestion as to the cause of the Rontgen 
rays, although exciting one's admiration by its clear con- 
sistency, does not lead to the desired end ; and therefore I 
will try to follow out the premisses of this paper to such 
conclusions as may relate to phenomena like those of the 
Rontgen rays. 

To the electrons we have assigned inertia and size, and we 
must therefore ascribe to them shape ; but a general con- 
ception of shape involves also the notion of deformability, 
which, therefore, we must consider as a possible property of 
the electron. The electron is therefore to be supposed 
capable of emitting vibrations due to the relative motions of 
its parts ; as light is supposed to be due to the motion of 
electrons as wholes, we see that the internal vibrations 
of electrons will have this much in common with light, that 
they are transmitted by the same sether, but they need have 
nothing else in common. We propose, then, to identify the 
Rontgen rays with these internal vibrations of our electrons. 
It might be expected that the electron, in executing the 
motions which cause light, would get strained and thrown 
into internal vibration, so that Rontgen rays would accom- 
pany ordinary light ; but the fact that Rontgen rays cannot 
be detected in association with light shows that the motion 
of the electron occurs either so that it is very free from shock 
and strain, or so that atoms promptly damp any internal 
vibrations of adjacent electrons. The way in which matter 
absorbs the energy of Rontgen rays shows that we may rely 
on atoms to suppress any small amount of Rontgen radiation 
that might tend to accompany ordinary light as emitted by 
electrons. Thus, then, appreciable Rontgen radiation is to be 
looked for only when free electrons are thrown into vigorous 
internal vibration. Now the encounter of an electron with 
an atom, in which it gives up a part of its kinetic energy to 
the atom as heat, is precisely the sort of action by which we 
should expect the electron to be thrown into internal vibration. 
Internal vibrations should originate where cathode or Lenard 

2 80 Mr. W. Sutherland on 

rays are absorbed, and most powerfully where the absorption 
is most powerful : this corresponds with all the facts as to 
the place of origin of Ronigen rays. 

As to what must be the order of magnitude of the length 
of the waves in the aether produced by the internal vibrations 
of the electrons, we can form no a priori estimate, but under 
the circumstances we are at liberty to assume that, like the 
size of the electron, it is small compared to that of atoms, and 
small also compared to molecular interspaces. We shall 
then have to do with systems of waves, which, when they 
fall on a body, can travel freely in the molecular interspaces, 
but are liable to absorption near the surfaces of molecules. 
The propagation of such a system of waves would take place 
almost entirely in the a?ther of the interspaces, as sound travels 
through a loose pile of stones mostly by the air-spaces ; the 
molecules cause absorption, but do not act as if they loaded 
the sether. Therefore when our system of waves enters a 
body it experiences no refraction. As to reflexion at the first 
layer of molecules which it encounters, we must remember 
that our wave-length is small compared to the radius of a 
molecule or atom, and that therefore in studying reflexion it 
suffices to study that from a single molecule ; whereas, with 
ordinary light, where the wave-length is large compared to 
atomic radius, we have to take the effect of a large number 
of contiguous molecules, if we are to reason out results com- 
parable with those observed in ordinary reflexion. Now the 
reflexion of our small waves from a single molecule will be of 
the same nature as reflexion from a sphere, and will be similar 
to diffuse scattering, a good deal of the scattering being 
towards the neighbouring interstices. Thus the attempt to 
reflect these waves from a material plane surface will be 
similar to that of attempting to reflect ordinary light from a 
large number of smooth spheres whose centres lie in a plane. 
If we take the average effect of a large number of molecules 
whose centres are by no means in a plane, as must be the 
case with our best reflecting surfaces, we see that a diffuse 
scattering of our small waves must take the place of re- 
flexion, and this is the experimental result with the Rontgen 

Any polarization that our system of waves might possess 
could not be detected by the ordinar}' optical appliances, 
because these depend on actions exercised by the molecules 
on the vibrations of light, whereas, as our small waves travel 
by the interstices between the molecules, their character is 
not controlled to any appreciable extent by the molecular 
structure. This result also agrees with the experimental one 

Cathode, Lenard, and Rontgen Rays, 281 

that polarization of the Rontgen rays cannot be detected by 
ordinary optical apparatus for the purpose. 

These negative properties have been explained chiefly by 
the assumed smallness of the wave-length, and have, there- 
fore, little direct connexion with our theory of the Rontgen 
rays beyond indicating the probability of a small wave-length 
for the Rontgen rays for similar reasons to those usually 
urged. We must, therefore, proceed to properties that our 
short waves must possess by virtue of their origin in vibrating 
electrons. In the first place, we should expect the electrons 
forming the neutrons in the aether to be set vibrating by our 
waves ; but if they produce no dissipation of the energy they 
will not cause any absorption, but will simply participate in the 
general asthereal operations of propagating the waves. But 
when the waves get amongst the electrons associated with 
atoms, and set them vibrating internally, there is called forth 
that resistance to the vibration which constitutes the damping 
action already spoken of. One of the probable results of such 
an action would be the setting of the acting and reacting 
atom and electron into relative motion, so causing the ab- 
sorbed Rontgen energy to appear as some form of radiant 
energy congenial to the atom and electron. In this way our 
waves could give rise to fluorescent and photographic effects 
in the manner of the Rontgen rays. If an electron absorbs 
enough of the energy of our small waves, it may be set into 
such vigorous motion as to escape from the atom with which 
it is acting and reacting, and appear as a free electron, or it 
may associate itself with an electron* to form an ion. At 
the foundation of our theory we suppose our small waves to 
be produced by the deformation of an electron during a 
vigorous transfer of energy from electron to atom ; and now 
we suppose this to be a reversible action, so that an electron 
set vibrating near to an atom can convert enough of its 
vibrational energy into translational kinetic energy to escape 
from the atom. With this legitimate dynamic assumption of 
reversibility, we can deduce from our hypothesis the produc- 
tion of free electrons or ions in a dielectric traversed by our 
small waves, which is in agreement with the remarkable pro- 
perty possessed by the Rontgen rays of making gases con- 
duct electricity well. The presence of scattered ions in a 
solid dielectric does not necessarily make it conduct. An 
experimental method of testing our theoretical conclusion, 
that Rontgen rays ought to have the same effect on solid 
dielectrics as on gases, would be to heat one till it gave de- 
cided signs of electrolytic conduction, and then test as to 

* [Atom ?] 

282 Mr. W. Sutherland on 

whether conductivity is increased by radiation with Rontgen 
rays. Experiments on liquid dielectrics should be easy enough. 
One of Rontgen's observations is of special importance. He 
found that air, through which Rontgen rays are passing, 
emits Rontgen rays ; and this is exactly what our theory 
would indicate, because, as we saw in discussing reflexion, 
each atom scatters our small waves as a reflecting sphere dis- 
tributes ordinary light. 

The remaining important positive facts concerning Rontgen 
rays relate to their absorption in passing through different 
substances. Our short waves in passing through a unit cube 
of substance in a direction parallel to one of the edges, while 
passing along the molecular interspaces, will be falling at 
intervals directly on opposing surfaces of atoms; and if n be 
the number of atoms per unit volume, and It the radius of 
each, the quantity of surface encountered by unit area of 
wave-front will be proportional to ?itR 2 , and the number 
of encounters in passing unit distance will be proportional to 
ni, so that as regards amount of encounter of wave-front with 
atoms the energy absorbed by the atoms will be proportional 
to 7?R 2 . But if the effectiveness of a collision in causing ab- 
sorption from a given area of wave-front in a given time is 
also proportional to the density of the matter in the atom, 
that is to m/R 3 , as we had to suppose in discussing the col- 
lisions of electrons and atoms, then the absorption of energy 
from our short waves in passage through unit length of 
different substances will be proportional to n?///R, that is to 
the density and inversely to atomic radius as with Lenard 
rays. The fact that Rontgen rays produce powerful fluores- 
cence in certain substances shows that there are special re- 
sonance phenomena that must be expected to produce decided 
variations in absorption from the simple form just discussed ; 
but the fact remains, as discovered by Rontgen, that by far 
the most important factor in the absorption of Rontgen rays 
is density. Benoist (Compt. Rend, cxxiv.) has found that 
the absorption of Rontgen rays by certain gases is propor- 
tional to the density, the factor of proportionality being 
nearly the same as for solids such as mica, phosphorus, and 
aluminium, though rising to a value six times as great in the 
case of platinum and palladium ; density is the prevailing, 
but not the only, property which determines the absorption 
of the Rontgen rays. But under different circumstances 
Rontgen-ray apparatus gives out rays of very different absorb- 
ability, or, as it is usually expressed, of different penetrative 
power. Thomson's theory of the Rontgen rays, as thin elec- 
tromagnetic pulses, does not seem to offer any feasible ex- 

Cathode, Lenard, and Rontgen Rays. 283 

plana tion of this fundamental fact. The theory of vibrating 
electrons requires that, in addition to the fundamental mode 
of vibration, we must contemplate a number of harmonics 
associated with it ; various combinations of fundamental and 
harmonics will be associated with different conditions of 
generation of the vibrations, and these will correspond to the 
Rontgen rays of different penetrative power. 

An interesting observation of Swinton's, that two colliding 
cathode streams do not give rise to Rontgen rays, is explained 
by our hypothesis, because the electrons are so small and so 
far apart that an appreciable number of collisions between 
the electrons of two colliding streams cannot occur. 

Some consequences of our line of reasoning, to which as 
yet no corresponding experimental results have been obtained, 
may now be indicated. The difference between cathode and 
anode is due to the fact that the attraction of metallic atoms 
for positive electrons is stronger than for negative ones, so 
that under a given electrical stress negative electrons break 
away as a cathode stream more easily than positive ones as 
an anode stream. But still, under strong enough electric 
stress at the anode, it ought to be possible to get an anode 
stream or anode rays similar to the cathode rays, but carry- 
ing positive electricity. These on encountering atoms, espe- 
cially the atoms of a solid body, should cause the emission of 
rays similar to the Rontgen, but possibly very different in 
detailed properties, such as wave-length. It is possible that 
the Becquerel rays may be examples of what we may call 
positive Rontgen rays, because, while we have seen that, in 
the majority of cases, electrons move relatively to atoms in 
the production of light, in such a manner that they do not 
experience shocks throwing them into internal vibration, the 
uranium atom may be so formed that it periodically collides 
with its satellite electron or electrons, in which case the atoms 
of uranium would be a source of radiation analogous to the 

According to our theory the velocity of the cathode stream 
is not a physical constant like the velocity of light through 
the aether, but ought to vary greatly according to the history 
of the stream, which starts with zero velocity and ends with 
the same. The velocity of the Rontgen rays should be of 
the order of that of light : we cannot assert that it should be 
exactly equal to that of light, because to waves of so short a 
length the neutrons may act as if they loaded the aether, so 
that Rontgen rays may suffer a refraction in aether in com- 
parison with light. The fact that the experimental velocities 
found for the cathode rays are of the order of the velocity of 

284 Dr. R. A. Lehfeldt on the 

light is a striking one, to be compared with the fact that in 
the vena contracta of a gas escaping from an orifice the 
maximum velocity attainable is nearly that of the agitation 
of the average molecule in the containing vessel or of sound 
in the gas. 

It appears as though a complete theory of electricity would 
be a kinetic theory, in which the place of the atoms or mole- 
cules of the kinetic theory of matter is taken by the electrons. 
The ion appears as a sort of molecule formed by the union of 
an atom or radical to an electron. But such large questions 
can hardly be opened up in the present connexion. We may 
summarize the contentions of the preceding pages in the two 
propositions : — 

The cathode and Lenard rays are streams, not of ions, but 
of free negative electrons. 

The Rontgen rays are caused by the internal vibrations of 
free electrons. 

Melbourne, Nov. 1898. 

XXI. Properties of Liquid Mixtures. — Part III.* Partially 
Miscible Liquids. By R. A. Lehfeldt, D.Sc] 

THE phenomena of complete mixture between two liquids, 
about which so little systematic knowledge is yet in 
existence, are connected with the phenomena of ordinary 
solution by an intermediate stage, that in which two liquids 
dissolve one another to a limited extent only. The study of 
such couples seems a promising field of investigation, on 
account of the intermediate position they occupy; it seems 
to offer the chance of extending some of the laws arrived at 
with regard to simple solution to the more complicated cases; 
I have therefore attempted to get some information on the 
equilibrium between incomplete mixtures and the vapour 
over them, and especially at the " critical point," i. e., the 
point at which incomplete miscibility passes over into com- 
plete. A recent short paper by Ostwald % draws attention 
to the importance of that point in the theory of mixtures. 

Choice of Liquids. 

The first point is to obtain suitable pairs of liquids for 
experiment. In order to study the properties of the critical 
point with ordinary vapour-pressure apparatus, it is necessary 

* Part I. Phil. Mag. (5) xl. p. 398; Part II. Phil. Mag. (5) xlvi. 
p. 46 ; reprinted, Proc. Phys. Soc. xvi. p. 83. 

t Communicated by the Physical Society : read Nov. 25, 1898. 
X Wied. Ann. lxiii. p. 336 (1897). 

Properties of Liquid Mixtures. 285 

that the pressure at the critical point should be below one 
atmosphere, and that limits very much the choice of liquids. 
As a rule, two liquids either mix completely cold, or if they 
do not do that, raising to the boiling-point does not suffice to 
make them mix ; two or three cases are all that I have been 
able to rind in which the point of complete mixture can be 
arrived at by boiling, and consequently corresponds to a 
vapour-pressure below the atmospheric. Many other pairs 
of incompletely miscible liquids have been studied by 
Alexejew and others, but to arrive at their critical points it 
was necessary to raise them to a high temperature in sealed 
tubes. A recent paper by Victor Rothmund * contains new 
observations on the relation between concentration and tem- 
perature, including the concentration and temperature of 
the critical point, made by Alexejew's method. That paper 
contains a long account of previous work on the subject, 
which makes it the less necessary for me to go over the same 
ground. I will therefore mention only what has been done 
on vapour-pressures, as Rothmund does not touch on that 
side of the subject, merely adding two remarks to his paper. 
First, it does not seem to have been noticed that normal 
organic liquids always mix completely : I hoped to have 
found a normal pair to study first, in order to avoid the com- 
plication due to the abnormality supposed to be molecular 
ao-o-regation in the liquid ; I have not succeeded in finding 
such a pair. All the incompletely miscible pairs of liquids 
so far noted include water, methyl alcohol, or a low fatty 
acid as one member. To those with accessible critical 
points mentioned by Rothmund, I have only one pair to add, 
viz., ethylene dibromide and formic acid ; these mix on 
boiling and separate into two layers when cold. I have not 
yet gone further with this couple ; the vapour-pressure 
observations below refer to the well-known cases of phenol 
and water, aniline and water. 

An account of previous experiments on the vapour-pressure 
of incompletely miscible liquids will indeed not take up much 
space, since, so far as I know, there is only one to record, 
viz., Konowalow's f measurements on isobutyl alcohol- water 
mixtures. His observations (made by a static method) give 
some points on the vapour-pressure curve up to 100° for 
(1) pure isobutyl alcohol (100 °/ ) : (2) mixtures containing 
94*05 °/o an d 6*17 °/o>both clear; (3) an undetermined mixture 
which separated into two layers. He unfortunately did not 
measure the solubility of the alcohol in water, or water in 

* Zeitschr.f.phys. Chem. xxvi. p. 433 (1898). 
f Wied. Ann. xiv. p. 43. 

Phil. Mag. S. 5. Vol. 47. No. 286. March 1899. X 

286 Dr. R. A. Lehfeldt on the 

the alcohol, at any of the temperatures for which vapour- 
pressure observations are recorded, so those data have to be 
supplied from Alexejew's results *. Konowalow, in the 
second part | of his paper, proceeds to show that the possible 
forms of curve showing vapour-pressure against concentration 
(temperature constant) are two : (i.) the flat part of the 
curve bounded by a rising portion at one end and a falling 
portion at the other ; (ii.) the flat part bounded by a falling 
portion at each end. Isobutyl alcohol- water mixtures give a 
curve of the latter kind. 

Isobutyl alcohol and water, however, possess a critical 
point at about 130°, i. e., much above the boiling-point of 
either. I therefore decided to study first mixtures of phenol 
and water, which become homogeneous in any proportions 
below 70°. 

The phenol was a commercial " pure " specimen ; to purify 
it further, it was placed in a distillation-flask and melted ; 
then air was drawn through it for about half an hour, whilst 
its temperature was kept at about 160° to 170°, in order to 
dry it. It w T as then distilled, and by far the larger part came 
over between 178° and 180°. The fraction collected between 
179 0, 5 and 180° (about half the mass) was used in the 
experiments. To make up mixtures, the process always 
adopted was to warm the stoppered bottle containing the 
phenol to just above the melting-point, and pour the required 
amount into a wei o-hing-flask. It was found that the moisture 
absorbed from the air during the process was quite inappre- 
ciable. The phenol, kept day after day at 40° to 50° ready 
for use, slowly turned pink, showing the presence of rosolic 
acid ; but a comparative colour-observation showed that the 
amount of impurity was probablv not more than 1/10,000. 
When it was necessary to estimate phenol in a mixture, that 
was done by the method of Koppeschaar, tribromphenol 
being formed and the excess of bromine replaced by iodine 
and titrated with thiosulphate. The method gave quite 
satisfactory results. 

Experimental Methods used. 

The measurements on phenol-mixtures gave results con- 
trary to my expectations, so that I became suspicious of the 
experimental methods. In the end I made use of four 
different kinds of apparatus, but found that they gave results 
in practical agreement, so that it became chiefly a question of 
convenience to decide between them. 

* Wied. Ann. xxviii. p. 315. f Wied. Ann. xiv. p. 222. 

Properties of Liquid Mixtures. 287 

The first method tried was the " dynamic," carried out 
with the same apparatus as described in Part II. It required 
no modification, except the use of a new thermometer, since 
the old one did not go above 60°. The new thermometer was 
a longer one, graduated in ^ from 0° to 100° (by C. E. 
Midler, ^No. 8). Its corrections were obtained in two ways : 
first by comparison with a standard (Reichsanstalt, 7347) at 
certain fixed temperatures, viz., the boiling-points of methyl 
acetate (57°), methyl alcohol (65°), and ethyl alcohol (78°); 
secondly, by measurements of the vapour-pressures of water 
under the same conditions as in the actual experiments ; in 
these conditions part of the stem w r as exposed. 

To use the apparatus the required mixture was weighed out 
from melted phenol and distilled water, then warmed up in 
the w T eighing-bottle until it became homogeneous, and poured 
into the tube of the vapour-pressure apparatus. The apparatus 
works satisfactorily except for mixtures on a very steep part 
of the curve of vapour-pressure (p) over concentration (z) ; 
w T hen dpjdz is great, the change of composition of the liquid, 
due to the evaporation becomes disproportionately important, 
and the static method is to be preferred ; in the case of phenol 
mixtures, however, that only affects a small part of the range 
— mixtures with 90 per cent, or more of phenol. 

The anomalous result that made me at first doubtful of the 
accuracy of the method was that up to 60 or 70 per cent, of 
phenol added to water made practically no difference to the 
vapour-pressure of the water. To check this, I made one or 
two experiments by the static method, in a barometer tube 
surrounded by an alcohol-vapour jacket of the usual pattern. 
They were not carried out with any attempt at accuracy, but 
sufficed to show that the previous observations could not be 
far wrong. The problem then was to determine the small 
difference in pressure between water and the phenol- water 
mixtures, and as for that purpose a differential gauge is 
obviously more appropriate, I set about designing and making 
the apparatus described below. Its design is based on a point 
of technique that does not seem to be much known, and to 
which, therefore, I should like to draw attention. If a glass 
tube be drawn out fine, sealed at one end, and evacuated, the 
sealed end may be broken under the surface of a liquid, 
which then flows in at any desired rate according to the 
diameter of the tube, and the tube may at any moment be 
fused off in the middle by a mouth blowpipe, without any 
inconvenience whatever. This process of filling with a liquid 
will I think be found advantageous in many cases. The only 
trouble about it is to get the canillary of the right bore"; 


288 Dr. R. A. Lehfeldt on the 

since the rate of flow depends on the fourth power of the 
radius it is easy to make the tube too wide or too narrow. 
Of course I made a good many failures at first, but after 
some practice could rely on getting the required condition. 
I used tubing of about 1 millim. internal diameter, and 
4 millim. external, and drew it out till the internal diameter 
was about one sixth of a millimetre ; a few centimetres of 
such a bore gives a convenient rate of flow for liquids of the 
viscosity of water. 

Ficr. 1. 



The apparatus for vapour-pressure measurements is shown 
in fig. 1. It consists of a U-tube, A, to serve as a gauge, 
carrying a branch, B, below, drawn out for filling as mentioned 
above. The top of the gauge-tube is bent round each side to 
the bulbs C, C, which are also provided with filling- tubes 

Properties of Liquid Mixtures. 289 

D, D'. The whole is shown flat in the diagram ; but as a 
matter of fact the side tubes C D and C D' were bent round 
till the bulbs nearly touched, to ensure their being of the same 
temperature. The apparatus was cleaned out with chromic 
acid, washed, and dried ; the capillaries were then drawn out 
and two of them sealed up, the third being left with the bit 
of wider tubing beyond the capillary untouched. By means 
of this it was attached to a mercury-pump, exhausted, and 
the capillary fused. The point B was then opened under 
mercury and fused off when the gauge contained sufficient : 
in the same way one of the bulbs was half filled with the 
mixture through D, arid then the other with water (which 
must, of course, be freed from air) through D' '. The apparatus, 
all of glass and hermetically sealed, is then ready for use : a 
glass millimetre scale is fastened with rubber bands to the 
gauge-tube, and it was immersed in a large glass jar of water. 
The scale was usually read by the telescope of a cathetometer 
and sometimes the screw micrometer of the telescope used 
to subdivide the graduation. The differential method avoids 
the necessity for any very great care in maintaining or mea- 
suring the temperature of the apparatus. It was found quite 
sufficient to heat the water-bath by leading a current of 
steam into it, and when the required temperature was 
reached, stop the steam for a moment and read the differ- 
ence of level. When the highest temperature (90°) was 
reached, some of the water was siphoned off, replaced by 
cold, the whole mass well stirred, and a reading taken. There 
was no noticeable lag in the indications of the gauge, the 
readings at the same temperature, rising and falling, being 
in good agreement. 

The fourth apparatus used was the Beckmann boiling-point 
apparatus, in its usual (second) form : with that observations 
at 100° were obtained of a kind to confirm the measurements 
made at somewhat lower temperatures with the vapour-? 
pressure apparatus. 

Observations of Vapour-pressure. 

The following observations were obtained with the differ-? 
ential pressure-gauge : t is the temper,: turo centigrade, p 
the vapour-pressure of the mixture, tt that of water, tt— p 
is therefore the difference observed with the gauge, and 
{it— p)/7r represents the relative lowering of the vapour- 
pressure of water by the addition of the quantity of phenol 

290 Dr. R. A. Lehfeldt on the 

Phenol-water Mixtures. 

67*36 per cent, of phenol. 

t 50° 65° 75° 85° 90° 

Tv-p 0-5 0-4 2-2 5-6 8-2 

(tt-^/tt 0-005 0-002 0-008 0-013 0-015 

77*82 per cent, of phenol. 

t 70° 75° 80° 85° 90° 

7T-p 2-6 5-0 8-2 12-5 17-1 

(tt-^/tt 0-011 0-017 0-023 0029 0032 

82*70 per cent, of phenol. 

t 40° 50° 60° 65° 70° 75° 80° 85° 90° 

7T-p < 1-4 5-6 111 14-8 19-2 25-2 32-2 42-3 539 

(jr-jp)/*-.. 0-025 0-061 0-075 0-079 0082 0-087 0-091 0098 0-103 

90*46 per cent, of phenol. 

t 25° 40° 50° 60° 65° 70° 75° 

tt-p 1-9 9-1 17-5 31-7 43-2 53-5 68 6 

(tt-^V'tt... 0-081 0-165 0-190 0213 0-231 0-230 0-237 

A mixture containing 7*74 per cent, of phenol gave no 
certain indication of a difference of pressure between the 
mixture and pure water. On this point, however, more 
reliable information is to be obtained with the Beckmann 
apparatus. It will be noticed from the preceding figures 
that the influence of the dissolved phenol becomes steadily 
greater as the temperature rises, e.g. 82 per cent, of phenol 
produces nearly twice as great a change of vapour-pressure 
proportionally at 90° as it does at 50°. In agreement with 
this the rise of vapour-pressures in dilute solutions of phenol 
is more marked at 100° than at the lower temperatures at 
which the vapour-pressure apparatus is available.- The result 
of an experiment on the boiling-point is as follows : — 

Per cent, 

Fall of 

rise of vapour- 
pressure, p — 7T. 







The general character of the results is sufficiently shown 

Properties of Liquid Mixtures. 


by fig. 2, in which the isothermals of 90° and 75° are suffi- 
ciently represented. That of 90° is comparable with the curve 






of Phenol- Water Mixtures. 






•7- i 


/ 3 


<* — r 












, ^ 










■ ■ 




300 B 





Per cent, phenol in water. 

for alcohol-toluene mixtures (see fig. 2 in the preceding 
memoir), only that the flatness extending over a great part of 
the range of concentrations is exaggerated in the phenol- water 
mixtures. The curve for 75° — still above the critical point — 
is still flatter; indeed it is imposible to say whether it rises or 
falls. Probably, therefore, below the critical point (where 
the vapour-pressure of phenol is inconveniently small for 
measurement) the isothermal, instead of consisting of a hori- 
zontal line bounded by two curves, would consist of a hori- 
zontal passing through the point representing pure water, 
bounded at the other end only by a descending curve. Such 


Dr. R. A. Lehfeldt on the 

an isothermal — that of 50° — is shown in fig. 2. The hori- 
zontal part ends at the point A (63 per cent.), beyond which 
the mixtures are homogeneous : the curve beyond A may 
theoretically meet the horizontal line at a finite angle, but 
that is certainly not distinguishable on the diagram. The 
curve is in fact exactly similar, so far as the experiments 
show, to that for 75°, and the pressure of the critical point, C 
on the diagram, which lies between them (at 68°*4) appears to 
make no difference whatever in this case— a case of great 
disparity in the vapour-pressures of the two components. 

For comparison,, a few experiments were made with aniline 
(not specially purified) and water. A mixture which consisted 
of two layers, even at the highest temperature used in the 
experiment, gave in the differential apparatus the following 
results : — 


7T — p 

(tt— p)U 







Whilst the Beckmann apparatus 

gave at 100° :— 

Per cent, 

Lowering of 


IT — p 

— ■ 













The second column gives the observed fall in temperature 
on adding aniline to the water ; the third column the rise of 
pressure corresponding, at the rate of 26*8 millim. per degree. 
Water at 100° is saturated by the addition of 6' 5 per cent, 
of aniline, and it will be seen that the vapour-pressure rises 
no further after that. The relative rise of vapour-pressure on 
saturation is 0*0324 at 100°, in satisfactory agreement with 
the numbers obtained by the differential gauge (0*0290, 
0*0 515, 0*0307), a tendency to increase with temperature 
being distinguishable here, as with phenol. Now suppose 
the vapour-pressure of a saturated mixture to be obtained in 
in this way : let the partial pressure of the water-vapour be 
that of pure water reduced by the normal amount (Raoult's 
law) due to the solution in it of the maximum quantity of 
aniline : and let the partial pressure of aniline-vapour be that 
of pure aniline reduced by the normal amount due to the 

Properties of Liquid Mixtures. 293 

solution in it of the maximum quantity of water. We get 
the following results at 100° : — ■ 

Vapour-pressure of water = 760 millim. 

Solubility of aniline = 6*5 per cent. = 1*32 molecular 
per cent. 

Partial pressure of water = 98*68 per cent, of 760 
= 7499 millim. 

Vapour-pressure of aniline = 46 millim.* 

Solubility of water in aniline = 8*7 per cent. = 33 mole- 
cular per cent. (Alexejew). 

Partial pressure of aniline = 30'8. 

Total pressure = 749*9 + 30*8 = 780*7. 

Observed pressure = 784*6 millim. 

The vapour-pressure of the saturated mixture is therefore 
given fairly well by the above rule. The rule cannot be 
applied to phenol mixtures, as below the critical point the 
vapour-pressure of phenol is too low to determine with 
accuracy. Konowalow's measurements of the vapour-pressure 
of isobutyl alcohol-water mixtures, combined with Alexejew's 
measurements of solubility, give the following results. At 

Vapour-pressure of water = 525 millim. 

Solubility of isobutyl alcohol = 7 per cent. = 1*8 mole- 
cular per cent. 

Partial pressure of water-vapour = 98*2 per cent, of 
525 = 515-5. 

Vapour-pressure of isobutyl alcohol = 378 millim. 

Solubility of water = 25 per cent. = 57*8 molecular per 

Partial pressure of isobutyl alcohol = 159*5. 

Sum = 675*1. 

Observed pressure =767. 

In this case the alcohol saturated with water contains more 
molecules of water than of alcohol, and it is not to be expected 
that the normal depression of the vapour-pressure should 
hold over so wide a range as 57*8 per cent. The numbers 
in fact show that the partial pressure of isobutyl alcohol 
must be very much greater — about 250 millim. The curve 
of partial pressures is therefore comparable with that for 
ethyl alcohol in benzene and toluene (see Part II. tables 
p. 53). 

* Kahlbaura, Zeitsch. f. phys. Chem. xxvi. p. 604. 


Dr. R. A. Lehfeldt on the 

Characteristic Surface for Phenol-Water Mixtures. 

To complete an account of the behaviour of phenol-water 
mixtures, it is necessary to draw a diagram of the relations 
between temperature and concentration ; this is given in 
fig. 3. Figs. 2 and 3 together, therefore, give a notion of the 
shape of the " characteristic surface," i. e. the surface showing 
the relations between concentration, temperature, andpressure. 
Fig. 2 contains three sections at right angles to the axis of 

































- is 











Per cent, phenol in water. 

temperature (for T = 90°, T = 75°, T=50° respectively), while 
fig. 3 gives one section at right angles to the axis of pressure 
(jo = l atmo). 

The behaviour of phenol-water mixtures is formally 

Properties of Liquid Mixtures. 295 

identical with that of benzoic acid and water*, but the curve 
brnnches of the diagram are of very different relative sizes to 
those of the last-named mixtures. The features of the diagram 
are as follows : — 

L. Freezing-point of water. 
0. Freezing-point of phenol. 
LS. Freezing-point of aqueous solutions of phenol. 
ONFGHMS. Freezing-point of solutions of water in phenol. 
S. Cryohydric point. 
MC. Saturation of water with phenol. 
NC. Saturation of (liquid) phenol with water. 
C. Critical point of mixture. 
The line LS is given by the thermodynamic equation 

0-02T 2 

where T is the absolute temperature of fusion of ice, L the 
latent heat of fusion, and t the resulting molecular depression 
of the freezing-point ; it accordingly starts with a slope of 
o, 2 for one per cent, of phenol. The initial slope of ON is 
given by a similar equation, and is 4°'15 for one per cent, of 
water ; a direct observation gave as a point of the curve 

80*5 per cent, phenol, 
melting-point + 5*0. 

This is marked with a dot in the figure, and lies to the right 
of ON ; by continuing the curve through the point so found 
until CN is met, we reach the point N where the phenol is 
saturated with water ; on increasing the concentration a 
second liquid layer appears, consisting of water saturated 
with phenol. NFGHM is purely hypothetical, referring to 
unstable mixtures ; actually any mixture of concentration 
between 8 per cent, and 77 per cent, of phenol will separate 
into two layers on cooling, and on further reduction of 
temperature freeze at the constant temperature (about -f 1 0, 5) 
represented by the horizontal straight line MN. The cryo- 
hydric point lies to the left of the saturation curve CM, so 
that it is actually attainable : its existence was shown by 
making a solution containing 5 '25 per cent, of phenol, and 
cooling it in a bath of ice and salt : it began to freeze with- 
out previous separation into two layers, and the temperature 
remained constant at — O- 9. About half of it was frozen, the 
beaker removed from the freezing-mixture, and some of the 
liquid remaining poured off for analysis ; it was found to 

* See van't Hoff, Vorlesungen ihber theoretische und physikalische 
Chemie, Heft i. p. 48. (Braunschweig, 1898.) 

296 Mr. W. B. Morton on the Propagation of 

contain 4*83 per cent, phenol : this concentration is therefore 
in equilibrium with both ice and solid phenol which had been 
deposited on the sides of the beaker. The cryohydric mixture 
therefore contains so little phenol that it may be looked upon 
as a dilute solution of phenol in water, and its calculated 
freezing-point, according to van't HofPs rule, would be — 1°'0, 
in agreement with the observed value — o, 9. Consequently 
solutions of strength between M and S will deposit phenol 
on cooling, those between L and S (0 to 4*83 per cent.) ice. 

The diagram is completed by the curve 5lCN which is 
drawn from Rothmund's observations*, which are indicated 
by dots ; my own observations (shown by crosses) are in 
practical agreement with bis and Alexejew's. 

Finally, the curves divide the diagram into regions, with 
the following meanings : — 

Below LS undercooled solutions of phenol, from which ice 
crystallizes out, with formation of the saturated solutions LS. 

Below SMNO supersaturated solutions of phenol, from 
which phenol crystallizes out with formation of the saturated 
solutions of phenol in water (SM) and water in phenol (NO). 

MONGM, unstable mixtures which separate into the two 
saturated solutions CM and CN, forming two liquid layers. 

Above LSMCNO homogeneous liquid mixtures. 
The Davy-Faraday Research Laboratory, Royal Institution, 
London, October, 1898. 

XXII. On the Propagation of Damped Electrical Oscillations 
along Parallel Wires. By W. B. Morton, M.A.\ 

IN a paper published in the Philosophical Magazine for 
►September 1898 Dr. E. H. Barton has compared the 
attenuation of electrical waves in their passage along parallel 
wires, as experimentally determined by him, with the formula 
given by Mr. Heaviside in his theory of long waves. The 
results show a large discrepancy between the theory and the 
experiments, the observed value of the attenuation constant 
being about twice too large. Dr. Barton discusses several 
possible causes of error and finds them inadequate, and 
suggests that the reason of the difference may lie either in 
(1) the nearness of the wires to one another, or (2) in the 
damping of the wave-train propagated by the oscillator. To 
these may be added (3) the consideration that the formulae 
used were deduced by Mr. Heaviside from the discussion of 
his " distortionless circuit," in which the matter is simplified 

* L. c. p. 452. 

t Communicated by the Physical Society : read Nov. 11, 1898. 

Damped Electrical Oscillations along Parallel Wires. 297 

by supposing sufficient leakage to counteract the distortion 
produced by the resistance of the leads, whereas in Dr. Barton's 
circuit the leakage was negligible. 

It is probable that the nearness of the wires has an appre- 
ciable effect on the phenomenon. The discrepancy would be 
diminished if the actual resistance of the wires was greater 
than that calculated by Dr. Barton from Lord Rayleigh's 
high-frequency formula. Now the effect of the neighbourhood 
of two wires carrying rapidly oscillating currents in opposite 
directions is to make the currents concentrate towards the 
inner sides of the wires*; and this would cause an increase 
in the effective resistance. 

I have examined the effects of (2) and (3), viz. of the 
damping and the want of balance in the constants of the 
circuit. The investigation is perhaps of some interest owing 
to the fact that these elements are always present in the 
ordinary experimental conditions ; although, as will be seen, 
we are led to the conclusion that in all actual cases their in- 
fluence on the phenomena is of quite negligible order. The 
method is the same as that used by Mr. Heaviside. 

General Theory. — Let the inductance of the circuit be L, its 
capacity 8, its resistance (of double wires) R, and its leakage- 
conductance K, all per unit length. An important part is 

played by the ratios j- and -^ ; we shall call these p and cr. 

When p and a are equal we have the " distortionless " circuit 
above referred to. 

Now if V be the difference of potential between the wires 
and C the current in the positive wire, we have the equations 

-S=( R+I 4) C > « 

-§=(*<>> w 


since LSt> 2 =^l, where v is the velocity of radiation. 
To simplify the algebra we shall work first withV= Y Q e~ mz+ni . 

* Cf. J. J. Thomson, ' Recent Researches/ p. 511. 

298 Mr. W. B. Morton on the Propagation of 

which can be made to represent damped periodic vibrations 
by giving complex values to m and n. 
The equation (3) now becomes 

mV=(/D + ra)(o- + 7i)s (4) 

and the connexion between C and V is given by (1), viz. 

L -R^Ln ^ 

To find the effect of a pure resistance B/ between the ends 
of the wires, as in Dr. Barton's experiments, put Y 1? Y 2 , 
G 1} G 2 for the potentials and currents in the incident and 
reflected waves respectively. Then we have 

p _ mY-i n _ -mVo 

B + W z R+hn 3 

also the total potential-difference V x + Y 2 is connected with 
total current G 1 -i-G 2 through resistance B' by Ohm's law, 

Vj+V^B^Ci + Co). 

These equations give for the reflexion factor 

Vj _ B + L?z-7»B / 

Y x B + Lw + mB/ (6; 

If the circuit be distortionless and B/ = Lv, then, as 
Mr. Heaviside showed, the absorption of the waves by the 
terminal resistance will be complete. We may regard this 
as the critical resistance for the circuit, and we shall express 
B' in terms of it by putting B'^^Lv. We then have 

Y 2 _ _ p + n — mvx 
Y l p + n + mvx 

Damped Wave-Train. — To pass to the case of a damped 
train transmitted from the origin in the positive direction of z 
we put 

in=—j3 + ia } n=—q + ip. 

The difference of potential between the wires at any point 
after the head of the wave-train has reached this point is then 
represented by an expression of the form 

Vo^-a'sin {pt—az). 

The velocity of propagation is -, the frequency -£—, the 

2-7777 a 

logarithmic decrement — -. If the waves suffered no attenu- 


Damped Electrical Oscillations along Parallel Wires. 299 
ation in their passage along the leads we should have 

j3z-gt = when z= P - 3 i. e. /3= ^. 

2 a, p 

In general, it is plain that ( ft ) measures the attenuation. 

Inserting the complex variables in equation (-1; we have 
v z (ft-ia) 2 =:{p-q^ip){cr — q + ip); 
... v ^- a * )=( f-f- q[p + (T)+p(7 . .... (8) 


2v 2 aft-=p{2q-p-a); (9) 


vW + **)= >/{p* + (q-p)*\{p a +{q-<rr\. . (10) 

Velocity of Propagation and Attenuation. — In actual cases 
p and cr are small compared with p. If the damping is con- 
siderable, q may be comparable with p. Accordingly we 
expand the right-hand side of (10) in ascending powers of p 
and a and solve for va and vft. As far as terms of the third 
order in p and cr we find • 

^f^ + a ^g#+ cd 

Hence the velocity of propagation 
and the attenuation 

= <7" o- 1 P +Q '-i- y^" 0- ) 8 + (% 2 -y)(p+^)(p-Q-) 2 , i m 

If p = <7, there is no distortion, the velocity of propagation 

is v, and the attenuation is ^— or -~- for all frequencies : and 
7 Ly bv l 

the damping has no effect on these quantities. 

We have an interesting particular case when p = cr = q. 

Then /3 = 0, and the state of affairs is given by 

Y =Y e-* t sin (pt— ~\ 

Here the damping and the attenuation are balanced, so that 

300 Mr. W. B. Morton on the Propagation of 

the wave -train in the wires is at any insfcant^u^Zy simple har- 
monic throughout. 

Numerical Values. — To obtain an estimate of the import- 
ance of the small terms of (13) and (14) I shall take the 
numbers given by Dr. Barton in his last paper (loc. cit.). 
Judging from his diagram (fig. 2) in that paper, the ampli- 
tude of the second positive maximum of the wave-train is 
about half that of the first. This would give 

e p =2 or q—^-~p^ — 

We have roughly 

p = 2tt x 35 x 10 6 = 22 x 10 7 , q = 2± x 10 6 , 
B = 69-5xl0 5 , L=19; 
.-. ^ = 37x10*, and er = 0. 
These values give for the velocity of propagation 
v{l --00000035} , 
and for the attenuation 


so that the corrections are quite negligible. We see from 
the expressions (13), (14) that the damping q only affects 
the value of the small terms introduced by the inequality of 
p and a. 

Effect of a Terminal Resistance. — To find the effect on the 
incident waves of a resistance (without inductance) inserted 
between the ends of the wires, we put in the complex values 
in the expression (7) for the reflexion-factor. We then get 

V 2 _ (p — q + vfix) +i(p — V*oc) 
Vi (—p + q + v/3x)—i(p + vax) 

=f+ig, say. 

Therefore to an incident wave e ipt corresponds a reflected 
wave {f+ig)e i P t \ or, taking real parts, with incident cos/?£ we 
have reflected 

/cos pt—g §\npt = */f 2 +y i cos (pt + 6), 

tan<9 = ^,; 

Damped Electrical Oscillations along Parallel Wires. 301 

so that the change of amplitude is accompanied by a change 
of phase. 

The values come out 

, 2 2 _ (p-g + vjBxY+ip-vax)* 
J ~ ty " (-p + q + v&v^+ip + vax)* ^7 

x*v 2 (cc* + j3*) + 2xv{/3(q-p)+*p} + (q- P y+p* 
g_ = 2xv{Pp-a{q-p)} 
f xy*{a* + l&)-.(q--p)*--p* " 



In order that there should be complete absorption of the 
incident waves it is necessary that the two squares in the 
numerator of (15) should vanish separately. This requires 

va vB 

— = — - — = X, say. 

p q-p ' J 

If we substitute va=:p\ and v/3=(q — p)\ in equations (8) 
and (9), and eliminate \ by division, we find the condition 
reduces to 

(p-<r){(q- P y+p*}=0, .-. p=<r. 

Therefore complete absorption is only attainable in the 
distortionless case. In general we can only reduce the 
reflected amplitude to a minimum. 

We can write (16) and (17) in the forms 

/2 2 _ ax* — 2hx + b 

tanfl=|= W ^~ A2 (19) 

From (18) we see that to any value of reflected amplitude 
correspond two values of the terminal resistance, say & l9 a? 2 . 
We can show that the corresponding phase-differences 1? 2 
are supplementary. 

For from (18) we have 




.: tan *, + tan *i=2 V«&-A«[,£^ + ^TTjJ 

— _Q 1 + Ay)(a3? 1 ,r 2 -& ) _ n 
=2V«6-A \ a tf-b)(a.*i-b) ~ U - 
Phil. Mag. S. 5. Vol. 47. No. 286. March 1899. Y 

302 Lord Kelvin on the Application of Sellmeier's Dynamical 
The minimum reflected amplitude is got when 

ffl = # 2 = \/ — 

V a 

The reflexion-factor is then — , and the phase-difference 

s/ab + h 

is ^-. When #=0 we have complete reflexion with unaltered 

phase; with # = co, or the circuit open, we have complete 
reflexion with reversed phase. The simultaneous alteration 
of amplitude and phase difference brings it about that we 
appear to pass continuously from amplitude +1 (x = 0) to 
amplitude —1 (<^ = oo ) without passing through amplitude zero. 
This apparent anomaly was pointed out to me by Dr. Barton. 
Putting in the values of a, a, A, and substituting for va and 
v/3 approximate values from (11), (12), we And that the 
minimum value of the reflexion-factor s/f 2 -\-g l is 

4:{p 2 + q 2 )\ 
and that the corresponding value of x is 

1 _ q{p-(r) 

neglecting higher terms in pa. 

Numerical Values, — Again using Dr. Barton's numbers we 
get for the minimum reflexion-factor the value *0004, and 
for the corresponding terminal resistance Lv(\ + '00009). 
If, therefore, the terminal resistance be adjusted until the 
reflected wave is a minimum, we may, without sensible error, 
take this resistance to be Lv, and ignore the reflected train 

Queen's College, Belfast, 
13th October, 1898. 

XXIII. Application of Sellmeier's Dynamical Theory to the 
Dark Lines D l3 D 2 produced by Sodium- Vapour, By 
Lord Kelvin, G, C. V. 0„ P.R.S.K* 

§ 1. T^OR a perfectly definite mechanical representation of 
J? Sellmeier's theory, imagine for each molecule of 
sodium-vapour a spherical hollow in ether, lined with a thin 
rigid spherical shell, of mass equal to the mass of homo- 
geneous ether which would fill the hollow. This rigid lining 

* Communicated by the Author, having been read before the Eoval 
Society of Edinburgh on Feb. 6, 1899. J 

Theory to Dark Lines D 1? D 2 p reduced by Sodlu m- Vapour. 303 

of the hollow we shall call the sheath of the molecule, or briefly 
the sheath. Within this put two rigid spherical shells, one 
inside the other, each movable and each repelled from the 
sheath with forces, or distribution of force, such that the 
centre of each is attracted towards the centre of the hollow 
with a force varying directly as the distance. These suppo- 
sitions merely put two of Sellmeier's single-atom vibrators 
into one sheath. 

§ 2. Imagine now a vast number of these diatomic molecules, 
equal and similar in every respect, to be distributed homo- 
geneously through all the ether which we have to consider as 
containing sodium-vapour. In the first place, let the density 
of the vapour be so small that the distance between nearest 
centres is great in comparison with the diameter of each 
molecule. And in the first place also, let us consider light 
whose wave-length is very large in comparison with the 
distance from centre to centre of nearest molecules. Subject 
to these conditions we have (Sellmeier's formula) 

v,\2 wit 8 mf 


where m, m, denote the ratios of the sums of the masses of 
one and the other of the movable shells of the diatomic 
molecules in any large volume of ether, to the mass of un- 
disturbed ether filling the same volume; /c, /c / the periods of 
vibration of one and the other of the two movable shells of 
one molecule, on the supposition that the sheath is held 
fixed ; v e the velocity of light in pure undisturbed ether ; 
v s the velocity of light of period t in the sodium-vapour. 

§ 3. For sodium-vapour, according to the measurements 
of Rowland and Bell*, published in 1887 and 1888 (probably 
the most accurate hitherto made), the periods of light corre- 
sponding to the exceedingly fine dark lines D 1? D 2 of the 
solar spectrum are '589618 and '589022 of a michronf. The 
mean of these is so nearly one thousand times their difference 
that we may take 

* Rowland, Phil. Mag. 1887, first half-year; Bell, Phil. Mag. 1888, 
first half-year. 

t " Michron " is the name which I have given to a special unit of 
time such that the velocity of light is one mikrom of space per michron of 
time, the mikrom being one millionth of a metre. The best determi- 
nations of the velocity of light in undisturbed ether give 300,000 kilometres, 
or 3 xlO'4 mikrom?, per second. This makes the michron £x 10- 14 of a 


304 Lord Kelvin '-on the Application ofSellmeier's Dynamical 
Hence if we put 

T =i(« + « y )(l+ i ^ 5 ) ..... (3); 

and if x be any numeric not exceeding 4 or 5 or 10, we 

V 2 -fc 2 ~ 2x+V T 2 -yc y 2_T 2¥-l * ' ' ^' 

Using this in (1), and denoting by //, the refractive index 
from ether to an ideal sodium-vapour with only the two 
disturbing atoms m, m n we find 

_ 1000 m 1000 m, ,.. 

2# + l 2#— 1 v J 


f 2 . 1000 t 2 . 1000 


§ 4. "When the period, and the corresponding value of x 
according to (3), is such as to make fi 2 negative, the light 
cannot enter the sodium-vapour. When the period is such as 
to make /jl 2 real, the proportion (according to Fresnel, and 
according to the most probable dynamics,) of normally incident 
light which enters the vapour is 

*- W (7). 

§ 5. Judging from the approximate equality in intensity 
of the bright lines D l3 D 2 of incandescent sodium-vapour ; 
and from the approximately equal strengths of the very fine 
dark lines D b D 2 of the solar spectrum ; and from the ap- 
proximately equal strengths, or equal breadths, of the dark 
lines !>!, D 2 observed in the analysis of the light of an incan- 
descent metal, or of the electric arc, seen through sodium- 
vapour of insufficient density to give much broadening of 
either line ; we see that m and m, cannot be very different, 
and we have as yet no experimental knowledge to show that 
either is greater than the other. I have therefore assumed 
them equal in the calculations and numerical illustrations 
described below. 

§ 6. At the beginning of the present year I had the great 
pleasure to receive from Professor Henri Becquerel, enclosed 
with a letter of date Dec. 31, 1898, two photographs of ano- 

Theory to Dark Lines D l5 D 2 produced by Sodium- Vapour. 305 

malous dispersion by prisms of sodium-vapour *, by which 1 
was astonished and delighted to see not merely a beautiful and 
perfect demonstration of the " anomalous dispersion " towards 
infinity on each side of the zero of refractivity, but also an 
illustration of the characteristic nullity of absorption and 
finite breadth of dark lines, originally shown in Sellmeier's 
formula t of 1872 and now, after 27 years, first actually 
seen. Each photograph showed dark spaces on the high sides 
of the D 1? D 2 lines, very narrow on one of the photographs; 
on the other much broader, and the one beside the D 2 nne 
decidedly broader than the one beside the D l line ; just as it 
should be according to Sellmeier's formula, according to 
which also the density of the vapour in the prism must have 
been greater in the latter case than in the former. Guessing 
from the ratio of the breadths of the dark bands to the space 
between their D„ D 2 borders, and from a slightly greater 
breadth of the one beside D 2 , I judged that m must in this 
case have been not very different from *0002 ; and I calculated 
accordingly from (6) the accompanying graphical represen- 
tation showing the value of 1 , represented by y in fig. 1. 

Fi g'- !• w=-0002. 








1 I 



















iym, t 












' -0 

3 \ 

n 1 



* A description of Professor Becquerel's experiments and results will 
be found in Comptes Rendus, Dec. 5, 1898, and Jan. 16, 1899. 

t Sellmeier, Pogg. Ann. vol. cxlv. (1872) pp. 399, 520 ; vol. cxlvii. 
(1872) pp. 387, 525. 

306 Lord Kelvin on the Application o/Sellmeiers Dynamical 

Fig. 2 represents similarly the value of 1 for m = '001, 

or density of vapour five times that in the case represented 
Fig.|2. ™=-00l. 


>r f 







1 ioco 





















> '' 

Fig. 3. 




999 D 

2 1000 C 

Fig. 4 

, 1001 












9919 Qj 10 

00 ( 

>, 10 




■■ X 

Fig. 5. 


11 1 1111 -o s r "T I I 1 

996 997 

999 D a 1000 °i 1001 1002 

by fig. 1. Figs. 3 and 4 represent the ratio of intensities of 
transmitted to normally incident light for the densities corre- 
sponding to figs. 1 and 2; and fig. 5 represents the ratio for 

Theory to Dark Lines D„ D 2 produced by Sodium- Vapour. 30? 

the density corresponding to the value m=*003. The fol- 
lowing table gives the breadths of the dark bands for densities 
of vapour corresponding to values of m from *0002 to fifteen 
times that value ; and fig. 6 represents graphically the breadths 
of the dark bands and their positions relatively to the bright 
lines D 1? D 2 for the first five values of m in the table. 

Values of m. 

Breadths of Bands. 

D r 

D 2 . 


















Fig. 6. 

VmV£S orM 


J D -° 



■ ,( 


5° 2 

_j d ' 

Pd 2 




1 -0 


_J D ' 



§ 7. According to Sellmeier's formula the light transmitted 
through a layer of sodium- vapour (or any transparent sub- 
stance to which the formula is applicable) is the same whatever 
be the thickness of the layer (provided of course that the 
thickness is at least several wave-lengths, and that the ordinary 
theory of the transmission of light through thin plates is 
taken into account when necessary). Thus the D 1? D 2 lines 
of the spectrum of solar light, which has travelled from the 
source through a hundred kilometres of sodium-vapour in the 
sun's atmosphere, must be identical in breadth and penumbras 
with those seen in a laboratory experiment in the spectrum of 

BOS Lord Rayleigh on the Cooling of Air by Radiation 

light transmitted through half a centimetre or a few centi- 
metres of sodium-vapour, of the same density as the densest 
part of the sodium-vapour in the portion of the solar atmo- 
sphere traversed by the light analysed in any particular 
observation. The question of temperature cannot occur except 
in so far as the density of the vapour, and the clustering in 
groups of atoms, or non-clustering (mist or vapour of sodium), 
are concerned. 

§ 8. A grand inference from the experimental foundation 
of Stokes'' and KirchhofPs original idea is that the periods 
of molecular vibration are the same to an exceedingly minute 
degree of accuracy through the great differences of range of 
vibration presented in the radiant molecules of an electric 
spark, electric arc, or flame, and in the molecules of a com- 
paratively cool vapour or gas giving dark lines in the spectrum 
of light transmitted through it. 

§ 9. It is much to be desired that laboratory experiments be 
made, notwithstanding their extreme difficulty, to determine 
the density and pressure of sodium-vapour through a wide 
range of temperature, and the relation between density, 
pressure, and temperature of gaseous sodium. 

XXIV. On the Cooling of Air by Radiation and Conduction, 
and on the Propagation of Sound. By Lord Rayleigh, 

ACCORDING to Laplace's theory of the propagation of 
Sound the expansions (and contractions) of the air 
are supposed to take place without transfer of heat. Many 
years ago Sir G. Stokes f discussed the question of the 
influence of radiation from the heated air upon the propagation 
of sound. He showed that such small radiating power as is 
admissible would tell rather upon the intensity than upon the 
velocity. If x be measured in the direction of propagation, 
the factor expressing the diminution of amplitude is e~ mx , 

m=2^1jL (1) 

In (1) 7 represents the ratio of specific heats (1*41), a is the 
velocity of sound, and q is such that e~^ represents the law of 
cooling by radiation of a small mass of air maintained at 
constant volume. If t denote the time required to traverse 
the distance x } r=-xja, and (1) may be taken to assert that 
the amplitude falls to any fraction, e. g. one-half, of its original 

* Communicated by the Author. 

t Phil. Mag. [4] i. p. 30o, 1851 : Theory of Sound, § 247. 


and Conduction, ami on the Propagation of Sound. 309 

value in 7 times the interval of time required by a mass of 
air to cool to the same fraction of its original excess of 
temperature. " There appear to be no data by which the 
latter interval can be fixed with any approach to precision ; 
but if we take it at one minute, the conclusion is that sound 
would be propagated for (seven) minutes, or travel over about 
(80) miles, without very serious loss from this cause 9> *. We 
shall presently return to the consideration of the probable 
value of q. 

Besides radiation there is also to be considered the influence 
of conductivity in causing transfer of heat, and further there 
are the effects of viscosity. The problems thus suggested 
have been solved by Stokes and Kirchhofff. If the law of 
propagation be 

u==e -m'x cos (nt—x/a), . . . . . (2) 

'=£{t^}' ^ 

in which the frequency of vibration is n/"27r, juJ is the kine- 
matic viscosity, and v the thermometric conductivity. In 
c.G.s. measure we may take /// = *14, v=\26, so that 

To take a particular case, let the frequency be 256 ; then 
since a = 33200, we find for the time of propagation during 
which the amplitude diminishes in the ratio of e : 1 , 

(m'a) =3560 seconds. 

Accordingly it is only very high sounds whose propaga- 
tion can be appreciably influenced by viscosity and conduc- 

If we combine the effects of radiation with those of viscosity 

and conduction, we have as the factor of attenuation 


g— (m+m^x 

where m + m , = '14:(q/a) + '12 (n?/a?) (4) 

In actual observations of sound we must expect the 
intensity to fall off in accordance with the law of inverse 
squares of distances. A very little experience of moderately 
distant sounds shows that in fact the intensity is in a high 
degree uncertain. These discrepancies are attributable to 

* Proc. Roy. Inst., April 9, 1879. 

t Pogg. Ann. vol. cxxxiv. p. 177, 1868 ; Theory of Sound, 2nd ed., § 348. 

310 Lord Rayleigh on the Cooling of Air by Radiation 

atmospheric refraction and reflexion, and they are sometimes 
very surprising. But the question remains whether in a 
uniform condition of the atmosphere the attenuation is sensibly 
more rapid than can be accounted for by the law of inverse 
squares. Some interesting experiments towards the elucida- 
tion of this matter have been published by Mr. Wilmer Duff *, 
who compared the distances of audibility of sounds proceeding 
respectively from two and from eight similar whistles. On an 
average the eight whistles were audible only about one-fourth 
further than a pair of whistles ; whereas, if the sphericity of 
the waves had been the only cause of attenuation, the dis- 
tances would have been as 2 to 1. Mr. Duff considers that in 
the circumstances of his experiments there was little oppor- 
tunity for atmospheric irregularities, and he attributes the 
greater part of the falling off to radiation. Calculating from 
(1) he deduces a radiating power such that a mass of air at 
any given excess of temperature above its surroundings will 
(if its volume remain constant) fall by radiation to one-half 
of that excess in about one-twelfth of a second. 

In this paper I propose to discuss further the question of the 
radiating power of air, and I shall contend that on various 
grounds it is necessary to restrict it to a value hundreds 
of twines smaller than that above mentioned. On this view 
Mr. Duff's results remain unexplained. For myself I should 
still be disposed to attribute them to atmospheric refraction. 
If further experiment should establish a rate of attenuation of 
the order in question as applicable in uniform air, it will I 
think be necessary to look for a cause not hitherto taken into 
account. We might imagine a delay in the equalization of 
the different sorts of energy in a gas undergoing compression, 
not wholly insensible in comparison with the time of vibra- 
tion of the sound. If in the dynamical theory we assimilate 
the molecules of a gas to hard smooth bodies which are nearly 
but not absolutely spherical, and trace the effect of a rapid 
compression, we see that at the first moment the increment 
of energy is wholly translational and thus produces a maxi- 
mum effect in opposing the. compression. A little later a due 
proportion of the excess of energy will have passed into, 
rotational forms which do not influence the pressure, and this 
will accordingly fall off. Any effect of the kind must give 
rise to dissipation, and the amount of it will increase with the 
time required for the transformations, i. e. in the above men- 
tioned illustration with the degree of approximation to the 
spherical form. In the case of absolute spheres no transforma- 
tion of translatory into rotatory energy, or vice versa, would 
* Phys. Keview, vol. vi. p. 129,.189B. 

and Conduction, and on the Propagation of Sound. 311 

occur in a finite time. There appears to be nothing in the 
behaviour of gases, as revealed to us by experiment, which 
forbids the supposition of a delay capable of influencing the 
propagation of sound. 

Returning now to the question of the radiating power of air, 
we may establish a sort of superior limit by an argument based 
upon the theory of exchanges, itself firmly established by the 
researches of B. Stewart. Consider a spherical mass of radius 
r, slightly and uniformly heated. Whatever may be the 
radiation proceeding from a unit of surface, it must be less 
than the radiation from an ideal black surface under the same 
conditions. Let us, however, suppose that the radiation is the 
same in both cases and inquire what would then be the rate 
of cooling. According to Bottomley* the emissivity of a 
blackened surface moderately heated is '0001. This is the 
amount of heat reckoned in water-gram-degree units emitted 
in one second from a square centimetre of surface heated 1° C. 
If the excess of temperature be 0, the whole emission is 

0x47n' 2 x-OOOl. 

On the other hand, the capacity for heat is 

7r>> 3 x-0013x-24, 

the first factor being the volume, the second the density^ and 
the third the specific heat of air referred as usual to water. 
Thus for the rate of cooling, 

dd -0003 1 
Jdt = ""•0013x«24xr = ~ r ^ nea * l 7> 
whence 0=0 Qe -V r , (5) 

O being the initial value of 6. The time in seconds of cooling 
in the ratio of e : 1 is thus represented numerically by r 
expressed in centims. 

When r is very great, the suppositions on which (5) is 
calculated will be approximately correct, and that equation 
will then represent the actual law of cooling of the sphere of 
air, supposed to be maintained uniform by mixing if neces- 
sary. But ordinary experience, and more especially the 
observations of Tyndall upon the diathermancy of air, would 
lead us to suppose that this condition of things would not be 
approached until r reached 1000 or perhaps 10,000 centims. 
For values of r comparable with the half wave-length of 
ordinary sounds, e.g. 30 centim., it would seem that the real 
time of cooling must be a large multiple of that given by (5) . 
* Everett, C.G.S, Units, 1891, p. 134. 

312 Lord Rayleigh on the Cooling of Air by Radiation 

At this rate the time of cooling of a mass of air must exceed, 
and probably largely exceed, 60 seconds. To suppose that 
this time is one-twelfth of a second would require a sphere of 
air 2 millim. in diameter to radiate as much heat as if it were 
of blackened copper at the same temperature. 

Although, if the above argument is correct, there seems 
little likelihood of the cooling of moderate masses of air being 
sensibly influenced by radiation, I thought it would be of 
interest to inquire whether the observed cooling (or heating) 
in an experiment on the lines of Clement and Desormes 
could be adequately explained by the conduction of heat from 
the walls of the vessel in accordance with the known con- 
ductivity of air. A nearly spherical vessel of glass of about 
35 centim. diameter, well encased, was fitted, air-tight, with two 
tubes. One of these led to a manometer charged with water or 
sulphuric acid ; the other was provided with a stopcock and 
connected with an air-pump. In making an experiment the 
stopcock was closed and a vacuum established in a limited 
volume upon the further side. A rapid opening and re- 
closing of the cock allowed a certain quantity of air to 
escape suddenly, and thus gave rise to a nearly uniform 
cooling of that remaining behind in the vessel. At the same 
moment the liquid rose in the manometer, and the obser- 
vation consisted in noting the times (given by a metronome 
beating seconds) at which the liquid in its descent passed 
the divisions of a scale, as the air recovered the temperature 
of the containing vessel. The first record would usually be 
at the third or fourth second from the turning of the cock, 
and the last after perhaps 120 seconds. In this way data 
are obtained for a plot of the curve of pressure; and the 
part actually observed has to be supplemented by extra- 
polation, so as to go back to the zero of time (the moment of 
turning the tap) and to allow for the drop which might occur 
subsequent to the last observation. An estimate, which 
cannot be much in error, is thus obtained of the w r hole rise in 
pressure during the recovery of temperature, and for the 
time, reckoned from the commencement, at which the rise 
is equal to one-half of the total. 

In some of the earlier experiments the w r hole rise of 
pressure (fall in the manometer) during the recovery of 
temperature was about 20 millim. of water, and the time of 
half recovery was 15 seconds. I was desirous of working 
with the minimum range, since only in this way could it be 
hoped to eliminate the effect of gravity, whereby the interior 
and still cool parts of the included air would be made to 
fall and so come into closer proximity to the walls, and thus 

and Conduction, and on the Propagation of Sound. 313 

accelerate the mean cooling. In order to diminish the dis- 
turbance due to capillarity, the bore of the manometer-tube, 
which stood in a large open cistern, was increased to about 
18 millim.*, and suitable optical arrangements were intro- 
duced to render small movements easily visible. By degrees 
the raDge was diminished, with a prolongation of the. time of 
half recovery to 18, 22, 24, and finally to about 26 seconds. 
The minimum range attained was represented by 3 or 4 
millim. of water, and at this stage there did not appear to be 
much further prolongation of cooling in progress. There 
seemed to be no appreciable difference whether the air was 
artificially dried or not, but in no case was the moisture 
sufficient to develop fog under the very small expansions 
employed. The result of the experiments maybe taken to be 
that when the influence of gravity was, as far as practicable, 
eliminated, the time of half recovery of temperature was 
about 26 seconds. 

It may perhaps be well to give an example of an actual 
experiment. Thus in one trial on Nov. 1, the recorded times 
of passage across the divisions of the scale were 3, 6, 11, 18, 
26, 35, 47, 67, 114 seconds. The divisions themselves were 
millimetres, but the actual movements of the meniscus were 
less in the proportion of about 2J : 1. A plot of these 
numbers shows that one division must be added to represent 
the movement between s and 3 s , and about as much 
for the movement to be expected between 114 s and oo . 
The whole range is thus 10 divisions (corresponding to 
4 millim. at the meniscus), and the mid point occurs at 
26 s . On each occasion 3 or 4 sets of readings were taken 
under given conditions with fairly accordant results. 

It now remains to compare with the time of heating 
derived from theory. The calculation is complicated by 
the consideration that wdien during the process any part 
becomes heated, it expands and compresses all the other 
parts, thereby developing heat in them. From the investi- 
gation which follows f, we see that the time of half recovery t 
is given by the formula 

' — -^r> < 6 ) 

in which a is the radius of the sphere, y the ratio of specific 
heats (1*41), and v is the thermometric conductivity, found 
by dividing the ordinary or calorimetric conductivity by the 

* It must not be forgotten that too large a diameter is objectionable, 
as leading to an augmentation of volume during an experiment, as the 
liquid falls. 

t See next paper, 

314 Lord "Rayleigh on Conduction of Heat in a Spherical 

thermal capacity of unit, volume. This thermal capacity is 
to be taken with volume constant, and it will be less than the 
thermal capacity with pressure constant in the ratio of 7 : 1. 
Accordingly v/y in (6) represents the latter thermal capacity, 
of which the experimental value is "00128 x *239, the first 
factor representing the density of air referred to water. 
Thus, if we take the calorimetric conductivity at '000056. we 
have in c.G.s. measure 

j>='258, v/ 7 =-183; 

and thence 

^ = -102« 2 . 

In the present apparatus a, determined by the contents, is 
16*4 centim., whence 

£=27*4 seconds. 

The agreement of the observed and calculated values is 
quite as close as could have been expected, and confirms the 
view that the transfer of heat is due to conduction, and that 
the part played by radiation is insensible. From a com- 
parison of the experimental and calculated curves, however, 
it seems probable that the effect of gravity was not wholly 
eliminated, and that the later stages of the phenomenon, at 
any rate, may still have been a little influenced by a downward 
movement of the central parts. 

XXY. On tlie Conduction of Heat in a Spherical Mass of Air 
confined by Walls at a Constant Temperature. By Lord 
Rayleigh, F.R.S.* 

IT is proposed to investigate the subsidence to thermal 
equilibrium of a gas slightly disturbed therefrom and 
included in a solid vessel whose walls retain a constant 
temperature. The problem differs from those considered by 
Fourier in consequence of the mobility of the gas, which 
may give rise to two kinds of complication. In the first 
place gravity, taking advantage of the different densities 
prevailing in various parts, tends to produce circulation. In 
many cases the subsidence to equilibrium must be greatly 
modified thereby. But this effect diminishes with the amount 
of the temperature disturbance, and for infinitesimal dis- 
turbances the influence of gravity disappears. On the other 
hand, the second complication remains, even though we limit 
ourselves to infinitesimal disturbances. When one part of 
the gas expands in consequence of reception of heat by 

* Communicated by the Author. 

Mass of Air confined by Walls at Constant Temperature, 315 

radiation or conduction, it compresses the remaining parts, 
and these in their turn become heated in accordance with the 
laws of gases. To take account of this effect a special 
investigation is necessary. 

But although the fixity of the boundary does not suffice to 
prevent local expansions and contractions and consequent 
motions of the gas, we may nevertheless neglect the inertia of 
these motions since they are very slow in comparison with 
the free oscillations of the mass regarded as a resonator. 
Accordingly the pressure, although variable with time, may 
be treated as uniform at any one moment throughout the mass. 

In the usual notation *, if .9 be the condensation and 6 the 
excess of temperature, the pressure p is given by 

p = kp(l + s + aO). . (1) 

The effect of a small sudden condensation s is to produce an 
elevation of temperature, which may be denoted by /3s. Let 
rfQ be the quantity of heat entering the element of volume in 
the time dt, measured by the rise of temperature which it 
would produce, if there were no " condensation/' Then 

d0 - R ds j. d $ i'9\ 

-Jt-Pd; + it* {Z) 

and, if the passage of dQ be the result of radiation and con- 
duction, we have 

§-**-&. .... • (3) 

In (3) v represents the " thermometric conductivity h found 
by dividing the conductivity by the thermal capacity of the 
gas (per unit volume), at constant volume. Its value for air 
at 0° and atmospheric pressure may be taken to be *26 
cm 2 ./sec. Also q represents the radiation, supposed to depend 
only upon the excess of temperature of the gas over that of 
the enclosure. 

If</Q=0, 0^/38, and in (1) 

p=A/){l + (l + ^}; 
so that 

i + «£=y, (4) 

where y is the well-known ratio of specific heats, whose value 
for air and several other gases is very nearly 1*41. 
In general from (2) and (3) 

i+4***'-* • • • • •?> 


'Theory of Sound,' §247. 

316 Lord Rayleigh on Conduction of Heat in a Spherical 

In order to find the normal modes into which the most 
general subsidence may be analysed, we are to assume that s 
and 6 are functions of the time solely through the factor e~ w . 
Since p is uniform, s-\-a0 must by (1) be of the form H e~ h \ 
where H is some constant ; so that if for brevity the factor 
e~ M be dropped, 

s + «<9=H; ....... (6) 

while from (5) 

vS7 2 0+{h-q)0 = hi3s (7) 

Eliminating s between (5) and (7), we get 

V s + m 2 (0-C)=.O, ..... (8) 

hy — q n 7</3H 

, ' 0-J^. • • • • 0) 

These equations are applicable in the general case, but when 
radiation and conduction are both operative the equation by 
which m is determined becomes rather complicated. If there 
be no conduction, v = 0. The solution is then very simple, 
and may be worth a moment's attention. 
Equations (6) and (7) give 

0= kpR , = (H)H, . . . (10) 

hy — q hy — q v ' 

Now the mean value of s throughout the mass, which does 
not change with the time, must be zero ; so that from (10) we 
obtain the alternatives 

(i.) h = q, (ii.) H = 0. 

Corresponding to (i.) we have with restoration of the time- 

0=(H/a>-««, s=0. .... (11) 

In this solution the temperature is uniform and the condensa- 
tion zero throughout the mass. By means of it any initial 
mean temperature may be provided for, so that in the 
remaining solutions the mean temperature may be considered 
to be zero. 

In the second alternative H = 0, so that s=—a6. Using 
this in (7) with v evanescent, we get 

(hy-q)d=0 (12) 

The second solution is accordingly 

0=<j>{x,y,z)e-«% s=-x<l>fay,z)e-Mr, . (13) 
where <£ denotes a function arbitrary throughout the mass, 
except for the restriction that its mean value must be zero. 

Mass of Air confined by Walls at Constant Temperature. 317 

Thus if <£) denote the initial value of 6 as a function of 
x, y, z, and O its mean value, the complete solution may be 

0=0 o «-8'+(0-0 o )<r-«'/v ] 

k . . . (14) 

s = « a (@_0 o ) £? -^/y J 


* + «0=a@ o <r-8< (15) 

It is on (15) that the variable part of the pressure depends. 

When the conductivity v is finite, the solutions are less 
simple and involve the form of the vessel in which the gas is 
contained. As a first example we may take the case of gas 
bounded by two parallel planes perpendicular to x, the 
temperature and condensation being even functions of x 
measured from the mid-plane. In this case \J" 2 = a n ldx 2 , and 
we get 

= C-f- Acosma?, -s/« = D + Acosw^, . . (16) 

S + a0 = «C-aD = H (17) 

By (9), (17) 

C-M!, D=i!LpM v . . . (18^ 

hy — q a [liy — q) 

There remain two conditions to be satisfied. The first is 
simply that = when x = + «, 2a being the distance between 
the walls. This gives 

C + Acos?na = (19) 

The remaining condition is given by the consideration that 
the mean value of s, proportional to \sdx, must vanish. 


ma.D+ sin ma. A = (20) 

From (18), (19), (20) we have as the equation for the 
admissible values of m, 

tan ma ctfiq — vm 2 

ma a(3{q + vm i y 

reducing for the case of evanescent q to 
tan ma 1 

ma aj3 

The general solution may be expressed in the series 

Phil. Mag. S. 5. Yol. 47. No 286. March 1899. 




318 Lord Rayleigh on Conduction of Heat in a Spherical 

where h ly h^ . . . are the values of 7i corresponding according 

to (9) with the various values of m, and ft, ft . . . are of the 


1 = cosm l x— cos m x a ~\ 

>>•••• (24) 
s 1 = — a(cos m±x — sin m^a/w^a) J 

It only remains to determine the arbitrary constants Af, 
A 2 , . t . to suit prescribed initial conditions. We will limit 
ourselves to the simpler case of q = 0, so that the values 
of m are given by (22). With use of this relation and 
putting for brevity a=l, we find from (24) 

ftft dx= 3— cos nil cos m 2 , 

a P 

1 j a ^ + l 
5 X 5 2 ax = 32 — cos m x cos m 2 ; 

'0 P 

so that 

( 1 2 dx + /3/a.\ sfrdv^O, .... (25) 

ft, ft being any (different) functions of the form (24). Also 

'^^^^{U.^}. . (26) 

There is now no difficulty in finding A 1? A 2 , . . . to suit 
arbitrary initial values of and its associated s, i. e. so that 

e=A 1 ft + A,0 2 + .. 






Thus to determine A 1? 

=A 1 ft + A a ft+ ... I 

= A 1 5 1 +A 2 5 2 + ... 

f 1 (® ft + pl'a . Ssj)^ = A 1 C (ft 2 + £/« . Sl 2 )dx 

+ A 2 (0 1 2 +l3/*.s 1 s 2 )da!+.. 


in which the coefficients of A 2 , A 3 . . . vanish by (25) ; so that 
by (26) 

M i+ ^1 = i^Jo 1(0<?i+/3a *- s ^- • (28) 

An important particular case is that in which © is constant, 
and accordingly S = 0. Since 

f 1 sin 7^ l + a/3 

1 V 1 dX= C0SWi = — ■ tt^-COSWi, 

Jo ™i «P 


Mass of Air confined by Walls at Constant Temperature. 3J9 

we have 

a __ 2® cos m x 
a/3 + cos 2 /^ 
For the pressure we have 

0+s/* = A 1 e-K(-cosm l +^^) + ..".". 

= 7j— cos m, . A x e- V + 

or in the particular case of (29), 

d + sa = 2® — 3^ 1 .- + (30) 

afj a/3 + COS 2 mi v ' 

If /3=0, we fall back upon a problem of the Fourier type. 
By (22) in that case 

ma = ^7r(l, 3, 5, . . .) 

cos 2 ma = a?fi 2 /m 2 a 2 , 
so that (30) becomes 

20 fe + ^v + --> • • • ^ 

or initially 

80/1 X , 1 . \ . _ 

^(p+ 35+52 + ...) »,«.e. 

The values of /* are given by 

/i= ^ (12 ' 32 ' 52 '--) ( 32 ) 

We will now pass on to the more important practical case 
of a spherical envelope of radius a. The equation (8) for 
(0—C) is identical with that which determines the vibrations 
of air * in a spherical case, and the solution may be expanded 
in Laplace's series. The typical term is 

d-(3={mr)-hJ nH (mr).Y n , . . . (33) 

Y n being the surface spherical harmonic of order n where 
n=0, 1, 2, 3 . . ., and J the symbol of Bessel's functions. In 
virtue of (6) we may as before equate — s/a — D, where D is 
another constant, to the right-hand member of (33) . The two 
conditions yet to be satisfied are that 0=0 when >*=a, and 
that the mean value of s throughout the sphere shall vanish, 

* l Theory of Sound/ vol. II. cli. xvii. 

320 Lord Rayleigh on Conduction of Heat in a Spherical . 

When the value of n is greater than zero, the first of these 
conditions gives C = and the second D = 0; so that 

6=-s/cc=(mr)-»J nH (mr) .Y n , . . . (34) 

and s + <x0 = O. Accordingly these terms contribute nothing 
to the pressure. It is further required that 

J n+i (ma)=0, (35) 

by which the admissible values of m are determined. The 

roots of (35) are discussed in ' Theory of Sound,' § 206 . . . ; 

but it is not necessary to go further into the matter here, as 

interest centres rather upon the case n = 0. 

. If we assume symmetry with respect to the centre of the 

1 d* 
sphere, we may replace V 2 in (8) by - y^ r, thus obtaining 

^|fr«" 0) +™Mfl-C)=0, . . . (36) 

of which the general solution is 

n fi . cos mr ^ sin mr 
u = \u -f A f- r> 

mr mr 

But for the present purpose the term r~ l cos mr is excluded, 
so that we may write 

= Q + B™2*, - S /«=D + B™', . (37) 
mr mr 


5 + a0 = a(C--D) = H. . . . {37 bis) 

The first special condition gives 

7waC + Bsinma = (38) 

The second, that the mean value of s shall vanish, gives on 

im 3 tt 3 D 4- B(sin ma — ma cos ma) = 0. . . (39) 

Equations (18), derived from (9) and (37 his), giving C 
and D in terms of H, hold good as before. Thus 

D _ g-h _ a/Bq-vm 9 - 

C ~ hafi ~ a/3{qivm 2 ) { } 

Equating this ratio to that derived from (38), (39), we find 

3 ma cos ma — sin ma _ vni 2 — aftq 
m?a* - sin ma a/3 (vm 2 + q) ' ^ ' 

Mass of Air confined by Walls at Constant Temperature. 321 

This is the equation from which m is to be found, after which 
h is given by (9). 

In the further discussion we will limit ourselves to the case 
°f 9 = ®> when (41) reduces to 


3a{3{mcotm-l), .... (42) 
in which a has been put equal to unity. Here by (40) 

Thus we may set as in (23), 

*-B^ + B,r-** + ) ■ m > 

s = B l e-^ t s 1 +B 3 «- V^ 2 + ) 

in which 

Q $\\\m x r sinra^ sin m,r 1 sin mi a ,... 

Vi= , Si=— a — —,(44) 

m x r m Y a m x r /3 m x a v ' 

and by (9) h l = vm 2 l /y. Also 

, , n 1 + a/3 sin m\a ,._,. 

S a + 1 = -^ l - (45) 

The process for determining B b B 2 , . . . . follows the same 
lines as before. By direct integration from (44) we find 

2W!W 2 ( l 

2 + /3/a . s^s 2 )r % dr 

_ sin (m, — m 2 ) sin (m 1 -f m 2 ) 2 sin m 1 sinw? 2 
mi—m 2 W]+m 2 3a/3 

a being put equal to unity. By means of equation (42) 
satisfied by m 1 and m 2 we may show that the quantity on the 
right in the above equation vanishes. For the sum of the 
-first two fractions is 

2??7 2 sin m x cos in 2 — 2m 3 sin m 2 cos m x 
?><, — w* 

of which the denominator by (42) is equal to 

Sa^{m l cot m 1 — m 2 cot m 2 ). 

ingly \\e& 


Accordingly {0& + P/* . s&ydr^Q. . . . (46) 

322 Lord Rayleigh on Conduction of Heat in a Spherical 
2V C l , a9 , ot » *j i sin 2wii , 2 sin 2 m x ( „. 

To determine the arbitrary constants B 1 . . . . from the given 
initial values of 6 and 5, say © and S, we proceed as usual. 
AVe limit ourselves to the term of zero order in spherical 
harmonics, i. e. to the supposition that 6, s are functions of r 
only. The terms of higher order in spherical harmonics, if 
present, are treated more easily, exactly as in the ordinary 
theory of the conduction of heat. By (43) 

6 = Bi#i + B 2 2 + i 

S=B l5l +B 2 5 2 + ... 

and thus 

P (e^ + ZS/a . $ Sl )r*dr= B x ( * (0* + 0/a . s^Alr 

Jo Jo 

+ B 2 \ l [6 A + PI a . S&) t*dr + ...., 


in which the coefficients of B 2 , B 3 , vanish by (46). The 

coefficient of Bx is given by (47) . Thus 

9rn 2 C 1 

I sin 2m a 2 sin 2 m l 

^ l \ 1 -~¥inT+~~^afi~ 

by which B} is determined. v ' 

An important particular case is that where © is constant 

and accordingly 8 vanishes. Now with use of (42) 

C l fl <2rf , _ Sm 1Ul ~~ ??l ! C0S 7Ul Sm 7??1 _ (1 "*" a fi) Sm ??l ! 

J l . " mj 3 3w l 3a^m l 

so that 

t, f . sin 2 m! , 2 sin 2 wit] 2j»i sin m. . @ ._. 

B '( 1 -^r + -3^-} : 3^ — •• ^ 

Bj, B 2 , .... being thus known, 6 and 5 are given as functions 
of the time and of the space coordinates by (43), (44). 
To determine the pressure in this case we have from (45) 

+ s/a _l + afiy^ sm 2 m.e- ht ,-.,. 


a$ jU &*£/ sin2m\' ' 

Mass of Air confined by Walls at Constant Temperature. 323 

the summation extending to all the values of m in (42). 
Since (for each term) the mean value of s is zero, the right- 
hand member of (51) represents also #/©, where 6 is the 
mean value of 0. 

If in (51) we suppose /3=0, we fall back upon a known 
Fourier solution, relative to the mean temperature of a 
spherical solid which having been initially at uniform tempe- 
rature ® throughout is afterwards maintained at zero all over 
the surface. From (42) we see that in this case sin m is small 
and of order ft. Approximately 

sin m=3aft/m ; 

and (51) reduces to 

of which by a known formula the right-hand member iden- 
tifies itself with unity when £ = 0. By (9) with restoration 
of a, 

/ i= (l 2 , 3 2 , 5 2 , ....>7r 2 /a 2 (53) 

In the general case we may obtain from (42) an approxi- 
mate value applicable when m is moderately large. The 
first approximation is m = iir, i denoting an integer. Suc- 
cessive operations give 

. t 3a ft 18* 2 /3 2 + 9* 3 /3 3 ,-.. 

™=^+ 1 - ^ • • (54) 

In like manner we find approximately in (51) 

sin 2 m (l+aft)/*ft _ 6(l + *g) f , _ 15*/3 + 9* 2 / 3 2 V 

i 2 7r 2 y 

.... (55) 

showing that the coefficients of the terms of high order in (51) 
differ from the corresponding terms in (52) only by the factor 
(1 + aft) or 7. 

In the numerical computation we take 7 = 1'41 , a/3 = '41. 
The series (54) suffices for finding m when i is greater than 2. 
The first two terms are found by trial and error with trigo- 
nometrical tables from (42) . In like manner the approximate 
value of the left-hand member of (51) therein given suffices 
when i is greater than 3. The results as far as / = 12 are 
recorded in the annexed table. 

. , 'daftf- sin 2mA 

324 On the Conduction of 'Heat in a Spherical Mass of Air. 




of (55). 





of (55). 












12 0104 








Thus the solution (51) of our problem is represented by 
^/e = -4942^ 1 -°" 4 ^' + -1799^ 20581 ^ , + . .. . . (56) 
where by (9) , with omission of g and restoration of a, 

t f lt = iT 2 v^a 2 (57) 

The numbers entered in the third column of the above 
table would add up to unity if continued far enough. The 
verification is best made by a comparison with the simpler 
series (52). If with t zero we call this series 2' and the 
present series 2, both 2 and 2' have unity for their sum, and 
accordingly yL' — 2 = 7 — 1, or 





Here 67/71- 2 = '8573, and the difference between this and the 
first term of 2, i. e. *4942, is "3631. The differences of the 
second, third, &c. terms are '0344, '0082, '0026, -0011, '0005, 
♦0000, &c, making a total of '4099. 






1 0000 

•60 1 









100 ! 





i 1 

We are now in a position to compute the right-hand 

Notices respecting New Books. 325 

member of (56) as a function of t f . The annexed table con- 
tains sufficient to give an idea of the course of the function. 
It is plotted in the figure. The second entry (t / = '05) requires 

the inclusion of 9 terms of the series. After t'='7 two terms 
suffice; and after t'='2'0 the first term represents the series 
to four places of decimals. 

By interpolation we find that the series attains the value *5 

^=•184. ...... (58) 

XXVI. Notices respecting New Books. 

An Elementary Course in the Integral Calculus. By Dr. D. A. 
Murray, Cornell University. Longmans, 1898. Pp. x + 288. 

r^E. MUEEAY states his object to be to present " the subjeet- 
-*^ matter, which is of an elementary character, in a simple 
manner." This he has succeeded in doing, and the work is well- 
arranged and the explanations given are exceedingly clear. In 
Chapter I. he treats Integration as a process of summation, and in 
Chapter II. as the inverse of differentiation. The author's object 
herein is to give the student a clear idea of what the Integral Calculi s 
is, and of the uses to which it may be applied. The first ten chapters 
are devoted to a treatment of the matters handled in such works as 
Williamson's, Edwards's, and other well-known treatises. Chap- 
ter XL treats of approximate integration, and the application of 
the Calculus to the measurement of areas. Here we have clear 
statements and proofs of the trapezoidal rule, Simpson's one- 
third rule, aud Duraud's rule. To this latter gentleman the 
author is indebted for valuable suggestions of use to engineering 
students. Prof. Durand has also put at Dr. Murray's disposal 
his article on " Integral Curves " (in the • Sibley Journal of 

826 Geological Society : — 

Engineering/ vol. xi. no. 4*), and his account of the ' Fundamental 
Theory of the Planimeter 'f . Chapter XIII. is devoted to ordinary 
differential equations. The Appendix (some 50 pages) discusses 
some of the matters in the text at greater length than is required 
by the elementary student, and also contains a large collection of 
the figures of the curves which are referred to in the exercises. 
Answers are given to these exercises, and further there is a full 
Index, and numerous brief historical notes which add to the 
utility of an excellent text-book. Dr. Murray suitably acknow- 
ledges his great indebtedness to his predecessors in the same field. 
The book is very neatly and correctly printed, and is of handy size. 

XXVII. Proceedings of Learned Societies. 


[Continued from vol. xlvi. p. 508.] 

November 9th, 1898.— W. Whitaker, B.A., F.E.S., President, 

in the Chair. 

rpHE following communications were read : — 

-*- 1. " On the Palaeozoic Eadiolarian Eocks of New South Wales.' 

By Prof. T. W. Edgeworth David, B.A., F.G.S., and E. F. Pittman, 

Esq., Assoc. R.S.M., Government Geologist, New South Wales. 

2. ' On the Radiolaria in the Devonian Eocks of New South 
Wales.' By G. J. Hinde, Ph.D., F.E.S., F.G.S. 

November 23rd.— W. Whitaker, B.A., F.E.S., President, 
in the Chair. 

The following communications were read : — 

1. ' Note on a Conglomerate near Melmerby (Cumberland;.' By 
J. E. Marr, Esq., MA., F.E.S., F.G.S. 

In this paper the author describes the occurrence of a con- 
glomeratic deposit which shows indubitable effects of earth- 
movement, not only on the included pebbles, bat also on the 
surface of one of the deposits. The rocks are coloured as basement 
Carboniferous rocks on the Geological Survey map. The Skiddaw 
Slates are succeeded by about 30 feet of a roughly stratified con- 
glomerate, followed by 20 to 30 feet of rock with small pebbles, and 
that by a second coarse conglomerate. The pebbles possess the 
outward form of glacial boulders, but many of them are slicken- 
sided, fractured, faulted, and indented. The striae are often 
curved, parallel, and covered by mineral deposit ; the grains of 
the matrix are embedded in the grooves, while slickensiding often 
occurs beneath the surface of the pebbles and the striae are seen 
to begin or end at a fault-plane. The surface of rock beneath 

. * Chapter XII. of the work before us is almost a reproduction of this 
article, as is also Appendix G which supplements the account in the text, 
t Here suitable reference is made to Prof. Henrici's Report (British 
Association, 1894) and to Prof. Hele Shaw's paper on 'Mechanical 
Integrators ' (' Proceedings of Institution of Civil Engineers, ' vol. lxxxii. 

The Geological Structure of the Southern Mdlverns, Sfc. 327 

the upper conglomerate was found to be slickensided. The way in 
which the surfaces of some of the pebbles have been squeezed-off 
suggests the possibility that their angular shape may be partially 
or wholly due to earth-movement. 

2. ' Geology of the Great Central Railway (New Extension to 
London of the Manchester, Sheffield & Lincolnshire Railway) : 
Rugby to Catesby.' By Beeby Thompson, Esq., F.G.S., F.C.S. 

In this paper the portion of the line, 10 miles in length, from 
Catesby to Rugby is described ; as the ground falls while the strata 
rise in this direction, quite low beds in the Lower Lias are met 
with near Rugby. The lowest zone exposed is that of Ammonites 
semicostatus, in the lower part of which, and in Boulder Clay derived 
from it, A. Turneri has been found. The next succeeding zone, that 
of A. ohtusus, although for the most part barren, yielded the charac- 
teristic fossils at its base. The o.vynotus-zone is well developed and 
well displayed, besides being richly fossiliferous. The zone of 
A. raricostatus merges into that of A. oxynotus below and that of 
A. armatus above, and is not more than 3 or 4 feet thick. The 
annatus-zone, beds between that and the Jameso?ii-zoTie, and the 
Jamesoni-zone itself follow ; the middle beds of the latter being 
rich in Rhynchonella and A. pettus, the name of this ammonite is 
attached to the zone bearing them. The Ibex-zone occurs east of 
Flecknoe, covered by rocks yielding A. Eenleyi ; and the highest 
beds of this cutting appear to belong to the capricornus-zone. 
Lists of the characteristic fossils of each zone are given, followed 
by a complete list of all those found in the Lower and Middle Lias 
of the cuttings, with a statement of their distribution. 

The Glacial deposits are described under the following headings : — 
Blue or local Boulder Clay, brown and grey contorted Boulder Clay, 
Chalky Lower Boulder Clay, (Mid- Glacial) sands and gravels, and red 
Upper Boulder Clay. 

The paper is accompanied by a measure 1 section along the 

3. ' On the Remains of Amia from Oligocene Strata in the Isle 
of Wight.' By E. T. Newton, Esq., E.R.S,, F.G.S. 

December 7th.— W. Whitaker, B.A., F.R.S., President; 
in the Chair. 

The following communication was read : — 

1. 'The Geological Structure of the Southern Malverns and 
of the adjacent District to the West.' By Prof. T. T. Groom, 
M.A., D.Sc, F.G.S. 

The Raggedstone and Midsummer Hills, consisting ess3ntially of 
massive gneissic and schistose rocks, are traversed by a curved 
depression which marks a line of profound dislocation, probably of 
the nature of a thrust-plane. This appears to dip towards the east, 
though with a relatively small hade. Along this depression occur 
strips of Cambrian and Silurian strata embedded in the Archaean 
massif, and indicating the presence of a deep and narrow dis- 
located synclinal fold. In places, the foliation of the schists 

328 Geological Society, 

shows a marked relation to the direction of this line, indicating 
in all probability a production of schists from the old material in 
post-Lower Palaeozoic times. 

The western boundary of the Archaean massif is everywhere a 
fault, apparently a thrust-plane, but with a small hade. The 
direction of this plane is in close relation with the axis of the over- 
fold into which the Cambrian rocks are thrown to the west of these 

The western boundary of Chase End Hill is likewise a fault, 
which is probably a thrust-plane with a tolerably low dip towards 
the eastern side. The thrust here also appears to have been 
accompanied by a secondary production of schists from the old 
gneissic series ; and the Cambrian strata are overthrown in the 
vicinity of the fault. 

There is no evidence for the overlap of the Cambrian Series sup- 
posed by Holl, the circumstance that the various zones of the 
Cambrian Series strike up against the Archaean axis being due to 

The Cambrian is represented by the following series : — 

; Upper Grey Shales. 
Coal Hill Igneous Band. 
Lower Grey Shales. 
Middle Igneous Band. 
; Upper Black Shales. 
Upper White-leaved-Oak Igneous Band. 
Lower Black Shales. 
Lower White-leaved-Oak Igneous Baud. 
2. Hollybusii Sandstone. 
i. hollybush quartzite and conglomerate. 

Possils are abundant in certain zones of each of the four sub- 
divisions of the series. 

The Grey Shales rest conformably on the Black Shales, but the 
mutual relations of the remaining subdivisions can be decided only 
by inference, the junctions being apparently everywhere faults. 
The junction between the Cambrian and Archaean is likewise a 

All four divisions of the Cambrian Series are invaded by small 
igneous bosses, laccolites, and intercalated sheets of diabase and 
andesitic basalt. These igneous rocks do not penetrate the May 
Hill Series. 

The May Hill Beds seem to rest with apparent conformity upon 
the Grey Shales, and do not transgress across the various Cambrian 
zones on to the Archaean in the manner hitherto supposed, the 
presumed outliers being small patches faulted into the Cambrian. 

The structure of the district is to be explained on the supposition 
that we are dealing with the western margin of an old mountain- 
chain overfolded towards the west ; the eastern portion of this 
range lies faulted down and buried beneath the Permian and 
Mesozoic of the Vale of Gloucester. All the characteristics of a 
folded chain are present, namely, the profound folds, overfolds, 
thrust-planes, and transverse faults ; and a typical Austonungs- 
zone is seen to the west. 

[ 329 ] 
XXVIII. Intelligence and Miscellaneous Articles. 



TN my second paper presented, last April, at the R. Istitufco 
-*- Yeneto, I dealt with the calorific phenomena observed in 
moistening pulverized bodies. The method of experiments has 
already been described f ; that is to say, of an arrangement whereby 
the liquid ascended to the powder, thoroughly dried, placed in a 
glass tube separated from it by a piece of light linen cloth. In 
this second series of experiments some modifications were intro- 
duced in order to keep the powder as dry as possible till the be- 
ginning of the experiment. The thermometer-bulb was placed in 
contact with the upper stratum of powder, it having been noticed that 
the increase of temperature was more marked from layer to layer. 

In the following tables are indicated some of the principal 
results obtained with pure charcoal and pure silica (Si0 2 ). 

Pure Charcoal (gr. 25). 

Name of the Liquid. 

Te »p. 
of the 

of the 


of the 





in cm. 3 

35 cm. 3 






Distilled water 

Absolute alcohol 

Sulphuric ether 



19 30 





9 20 




Acetic ether 


Bisulphide of carbon. 

Name of the Liquid. 

of the 

of the 

of the 


of the 

absorbed . 

Distilled water .. 

Absolute alcohol 

Sulphuric ether 


18 96 

19 49 

19 60 



37 cm. 3 





Acetic ether 


Bisulphide of carbon. . . 

1 made also many experiments in order to determine the number 
of calories produced by moistening powder. I adopted a special 

* " Intorno ad calore che si sviluppa nel bagnare le polveri. Nuove 
ricerche termometriche e calorimetriche." Atti del H. Istituto Yeneto, 
t. 9, serie vii. p. 927. 

t Atti del R. Istituto Veneto, t. 8, serie vii. p. 502; Phil. Mag. 
vol. xliv., August 1897. 


Intelligence and Miscellaneous Articles. 

calorimeter made of thin sheet brass, consisting of two cylindrical 
tubes, one within the other, having a common axis. The base of 
the internal tube was perforated to allow the escape of air, when 
pouring the liquid upon the powder with which the tube was 
filled. A flannel disk placed at the base prevented the escape of 
the powder. 

The powder was thoroughly dried before being poured into the 
internal tube, which was closed at the top with an indiarubber 
stopper, and suspended by three silk threads, within a vessel con- 
taining chloride of calcium. The external tube contained distilled 
water into which was immersed a delicate thermometer. 

Calories developed by Charcoal moistened with Distilled Water. 

Weight of the 

Volume of the 

Calories, !<*?££■* 

44 gr. 





60 cm. 3 





629-00 14-29 
569-80 14-25 
573-30 14-33 
514-30 14-69 
440-30 14-67 

Calories developed by Silica moistened with Distilled Water. 

Weight of the 

Volume of the 


Calories by 1 gr. 




of powder. 

50 gr. 

72 cm. 3 



























Meissner, in his experiments *, did not use a fixed weight of 
water ; at one time he would use a quantity of water equal in 
weight to the powder, at another time a quantity double, and at 
times much less. In my own case, however, I always used that 
quantity of liquid which I found would be absorbed, by capillarity, 
by a quantity of charcoal or silica equal to that contained in the 
calorimeter. Had I poured on the powder a smaller amount of 
water, parts would have remained umnoistened ; a larger quantity 
would have absorbed a part of the heat generated. 

The foregoing results may be of interest, not only to physicists 
in general, but to students of geothermic phenomena. In fact, 
the reader will find in my original pamphlet an account of certain 
experiments in which silica, moistened with a proportionate 
quantity of water, rose from an initial temperature of 19° to that 
of 70°. 

Venice, June 1898. 
* " Ueber diebeiin Benetzen purverfbrniiger Korper auftretecde Warme 
tonung." Wiedemann's Annalen, xix. (1886). 

Intelligence and Miscellaneous Articles. 331 



It was shown by Ampere * how a magnet can be made to rotate 

about its axis under the influence of a steady current, and Faraday f 

showed how the rotation of a magnet about its axis can give rise 

to a steady current +. 

By combining these two experiments we obtain a case of 
induction by steady currents : a steady primary current in the 

circuit E or E x gives rise to a steady secondary current in the 
circuit E x or E, a rotating magnet M forming the connecting link. 

In order to show the effect, a brake is applied to the magnet M, 
a battery is inserted into the circuit E or E 2 and a galvanometer 
into the other circuit E 3 or E. On closing the circuit we observe 
that the suspended system of the galvanometer remains at zero ; 
but, on removing the brake from the magnet, this begins to rotate 
and the galvanometer shows a deflection which increases con- 
tinuously until the magnet turns quite freely. A brake is desirable 
which admits of a graduated application. 

It appears to me that this experiment is worthy of notice in 
consideration of its illustrative character §. 

Degerloch (Wiirttemberg), 
December 12th, 1898. 

To the Editors of the Philosophical Magazine. 


I have read with considerable interest the paper by Dr. Cook 
on the " Brush Discharge " in the January number of your 
Magazine. May I be allowed to call attention to some experiments 
made by Lord Blythswood about two years ago, which are of a very 

* Recueil d? Observations, p. 177 (1821). Lettre a M. van Beck. 

t 'Experimental Researches/ series ii., §§ 217-230 (1832); see also 
series xxviii. (1851). 

X This phenomenon is generally known by the name of " unipolar '' 
induction ; " autopolar " induction (induction autopolaire, Gleichpolin- 
duction), it seems to me, would be better; and therefore 1 propose this 
term. Then, in contradistinction, " heteropolar " induction (induction 
he"teropolaire, Wechselpolinduction) can be used for signifying induction 
by both the poles alternately (dynamo &c). 

§ See Phil. Mag. vol, xlvi. p. 428 (Oct, 1808)., and p. o71 (Bee. 1898). 


Intelligence and Miscellaneous Articles. 

similar character. In this case, however, it was the negative glow 
which was the source of radiation. The machine used was of the 
Wimshurst type, having 160 3-ft. plates each carrying 16 sectors. 

The experiments were made to determine the effect of the rays 
from the negative g]ow on a photographic plate. 

Small metallic objects were " radiographed," being placed in 
front of, but not touching, a sensitized plate. The whole was then 
enclosed in a zinc box (with a small hole cut in the side facing the 
source of radiation), which was carefully earthed in order to 
prevent any charge on the metallic object affecting the plate. A. 
piece of 1 mil. aluminium-foil was then placed between the source 
of radiation and the plate, thus completing the metallic sheath. 
The arrangement is shown below. 

<--Z/nc Box 




"When these precautions were not observed, brush discharges 
took place from the points of the metallic object (generally a 
small wheel) which strongly affected the plate. 

Distinct shadows of the small wheel were obtained after 5 
minutes' exposure in a darkened room. It was at first thought that 
these photographs were produced by rays similar to the .r-rays, 
which had traversed the aluminium-foil ; but it appeared afterwards 
that the whole effect was apparently due to minute holes in the 
aluminium-foil, since, when the apparatus was wrapped in black 
velvet, no effects were produced on the plate. That such effects were 
obtained, however, seems to show that the negative glow possesses 
strong actinic power, as shown so conclusively by Dr. Cook. 

A possible explanation of the diminished effect observed by 
Dr. Cook on an Electroscope placed at a distance from a point at 
which a Brush discharge is taking place, when an induction-coil was 
used in place of the Wimshurst machine, seems to be that the 
electrification produced would depend on the R.M.S. potential- 
difference rather than on the maximum value, as indicated by the 
spark-length. Similarly for the mechanical force produced by the 
wind from the points. 

Tours very truly, 

Blythswood Laboratory, E. W. Maechajstt. 

Renfrew, N.B. 

Jan. 30, 1899. 





\b b a r p; 


APRIL 1899. 

XXIX. Longitudinal Vibrations in Solid and Hollow 
Cylinders. By C. Chree, Sc.D., LL.D., F.B.S* 


§ 1. TT1HE frequency kfiir of longitudinal vibrations in the 
A ideal isotropic bar of infinitely small cross section 
has long been known to be given by 

k=p K/Efp, (1) 

where p is the density, E Young's modulus ; p is given by 
<p = i7r/l when the ends are both free or both fixed 
p=(2i + l)7r/(2l) when one end is free, the other fixed, 
I being the length of the rod and i a positive integer. 

For the fundamental or lowest note 2 = 1. 
For a circular bar whose radius a, though small compared 
to I, is not wholly negligible, the closer approximation 

k=p(E/p)i{l-}pWa 2 \, (2) 

where tj is Poisson's ratio, was obtained independently by Prof. 
Pochhammer j" and Lord RayleighJ fully 20 years ago. 

§ 2. The subject has been treated by myself in three 
papers in the ' Quarterly Journal of . . . Mathematics ' (A) 
(p. 287, 1886), (B) (p. 317, 1889), (C) (p. 340, 1890). 

In (A) I arrived at (2) describing it (/. c. p. 296) as 

* Communicated by the Physical Society : read December 9, 1898. 

t Crelle, vol. lxxxi^. (1876). 

\ i Theory of Sound,' vol. i. art. 157. 

Phil. Mag. S. 5. Vol. 17. No. 287. April 1899. 2 A 

334 Dr. C. Chree on Longitudinal 

u obtained as a second approximation by Lord Eayleigh." I 
further said, " We do not think, however, that his proof affords 
any means of judging of the degree of accuracy of the result, 
as it is founded on a more or less probable hypothesis and 
does not profess to be rigid/' I subsequently learned that 
Lord Eayleigh did not admit any want of rigidity in his 
proof, and it appears without modification in the second 
edition of his Treatise on ' Sound/ I much regret having to 
differ from so eminent an authority, but I have not altered 
my original opinion. 

In (B) I reached the more general result 

k=p(V/p)i(l-i P YK 2 ), (3) 

where k is the radius of gyration of the cross section of the 
rod about its axis. This was established by a strict elastic 
solid method for an elliptic section, and in a somewhat less 
rigid way for a rectangular section. (A) and (B) were con- 
fined, like the investigations of Lord Eayleigh and Professor 
Pochhammer, to isotropic materials. 

In (C) I considered the more general case of an seolotropic 
bar whose long axis was an axis of material symmetry, and 
found by strict elastic solid methods that (2) still held for 
a circular section, if E denoted Young's modulus for stress 
along the length of the bar, and rj Poissou's ratio for the 
consequent perpendicular contractions. Further, applying 
Lord Eayleigh/ s method, modified in a way I deem necessary, 
I obtained (3) for any form of cross section. 

§ 3. Since the publication of (( J) Mr. Love has discussed the 
subject in vol. ii. of his l Treatise on Elasticity/ On his p. 1 19 * 
he refers to (2) as " first given by Prof. Pochhammer . . . 
and afterwards apparently independently by Mr. Chree/'' 
Again, in the new edition of his ' Sound ' Lord Eayleigh, 
after deducing (2), says a A more complete solution. . . has 
been given by Pochhammer ... A similar investigation has 
also been published by Chree.'"' 

In view of these remarks, I take this opportunity of stating 
explicitly : — 

1. That Pochhammer's work was wholly unknown to me 
until the appearance of Love's l Elasticity/ 

2. That my method of solution in (A) is essentially different 
from Pochhainmer's, while the methods in (B) and (C) are 
absolutely different from his. The method of (A) agrees 
with Pochhammer's in employing the equations of elasticity 
in cylindrical coordinates. After obtaining, however, — as is 

* The preface, p. 13 ; describes the result as " obtained independently " 
by me._ 

Vibrations in Solid and Hollow Cylinders. 335 

customary in most elastic problems — the differential equation 
for the dilatation, Pochhammer obtains a differential equation 
for the quantity 

du dw\ 

1 /du 


u and w being the displacements parallel to the radius r and 
the axis z, and uses this quantity as a stepping-stone to the 
values of u and w. On the other hand, I succeeded in 
separating u and w so as to obtain at once two differential 
equations, in one of which u appeared alone with the dilatation, 
while the other contained only w and the dilatation (see (28) 
and (29) later). 

§ 4. There are two other points in Mr. Love's Treatise to 
which I should like to refer. In his art. 263 he substitutes 
the term extensional for longitudinal, adding in explanation, 
" The vibrations here considered are the ' longitudinal ' vibra- 
tions of Lord Kayleigh's Theory of Sound. We have 
described them as l extensional/ to avoid the suggestion that 
there is no lateral motion of the parts of the rod." 

I am altogether in sympathy with the object which Mr. 
Love has in view (1 expressed myself somewhat strongly on 
the point in (A) p. 296, and (C) pp. 351-2), but I doubt the 
wisdom of attempting to displace a term so generally adopted 
as longitudinal. 

In the second matter I regret to find myself at variance 
with Mr. Love. Referring to transverse vibrations in a rod, 
he says on p. 124 of his vol. ii., " the boundary conditions at 
free ends cannot be satisfied exactly ... as they can in the . . . 
extensional (longitudinal) modes/'' In reality, however, the 
boundary equations at a free end are not exactly satisfied in 
the case of longitudinal vibrations either by Pochhammer's 
solution or my own. The slip may be a purely verbal one on 
Mr. Love's part, but his readers might be led to accept the 
statement as accurate owing to a slight error in the expression 

for the shearing stress zr near the top of Mr. Love's p. 120. 
We find there 

«r= 2^{ 7 A j- J (/c'r) + . . . }e«<7*+?0, 

where i= ^ — ±. 

The correct expression (compare Mr. Love's second boundary 
equation on p. 118) is 

^ = ^{2 7 A ^-J (ft'r) + . . .}«to # +*>. 

2 A2 

336 Dr. C. Chree on Longitudinal 

Owing to the omission of i in the expression on p. 120 it 

looks as if zr vanished for the same values of z as the normal 

stress zz. In reality, as I showed in (A) p. 295, zr does not 
vanish over a terminal free section of radius a, but is of the 
order r(r 2 — a 2 )/l 3 , where r is the perpendicular on the axis. 

We are quite justified in neglecting zr when terms of order 
(a/1) 3 are negligible, but strictly speaking the solution is so 
far only an approximate one when the ends are free. 

A New Method. 

§ 5. In the Camb. Phil. Soc. Trans, vol. xv. pp. 313-337 
I showed how the mean values of the strains and stresses 
might be obtained in any elastic solid problem independently 
of a complete solution. For isotropic materials I obtained 
(/. c. p. 318) three formulas of the type 

E \\\ d -ldxdydz= N[ {I*-ri£Kw'+Yy))dx dy dz 

■i;(Fff + (fy)}dS, . (4) 


where a, fi, y are components of displacement, X, Y, Z of 
bodily forces, and F, G, H of surface forces. The volume 
integrals extend throughout the entire volume, and the surface 
integrals over the entire surface of the solid. As was explicitly 
stated in proving the results (I. c. p. 315), X, Y, Z may include 
' reversed effective forces ' 

dPa d 2 /3 d 2 ry 

p dt 2 ' p dF' - p W 

where p is the density. 

In the present application there are no real bodily forces ; 
we may also leave surface forces out of account, if we suppose 
that when one end of a rod is held, that end lies in the plane 
2 = 0. F 

Supposing the rod to vibrate with frequency &/27T, we have 

so that we replace (4) by 

"^JJJ ^ € *^^ €fe= ^^jHU ^^ — ^C«a?+ifiEy)l^»^«fc, . (5) 

and similarly with the two other equations of the same type. 
I he only other result required is one established in my 

Vibrations in Solid and Hollow Cylinders. 337 

paper (B) ; viz., that the general solution of the elastic solid 
equations of motion in which the terms contain cos pz or 
s'mpz consists of two independent parts. In the first, which 
alone applies to longitudinal vibrations, a. and j3 are odd and 
7 an even function of x and y. 
We may thus assume 

a = cos kt cos (pz — e) { A^ 4- A/y + A 3 # 3 4- A 3 'x*y 

+a b 'v+a, ,, v +...}, 

/3=cos^cos(^.~-e){B 1 A< + B/# + B 3 ^ 3 + ...}, I * W 

7 = cos kt sin (pz - e) {C + C 2 x 2 + QJxy + C 2 ff f + •••}> ) 

where e is a constant depending on the position of the origin 
of coordinates and the terminal conditions. It is obvious 
from various considerations that the same e occurs in the 
values of a, /3, y. Certain relations must subsist between the 
constants A, B, C in the above expressions, in virtue of the 
body- stress equations, but we do not require to take any heed 
of these for our present purpose. 

§ 6. As the validity of solutions in series has been a subject 
of contention in other elastic solid problems, some doubt may 
be entertained as to the results (6). I would be the last to 
deny the reasonableness of this, because I do not myself 
regard (6) as universally applicable. 

According to my investigations, quantities such as 
(A 3 # 3 /A.i#) are of the order (greatest diameter/nodal length) 2 , 
and the series become less rapidly convergent as (greatest 
diameter/nodal length) increases. In other words, increase 
either in (greatest diameter/rod length) or in the order of the 
" harmonic " of the fundamental note reduces the rapidity of 
convergence. The proper interpretation, however, to put on 
this is not that (6) is a wrong formula for longitudinal 
vibrations, but simply that under the conditions specified the 
vibrations tend to depart too widely from the longitudinal 
type. If we apply this solution the results deduced from it 
themselves tend to show the degree of rapidity of the con- 
vergence, and what we have to do is to keep our eye on the 
results and accept them only so long as they are consistent 
with rapid convergency. 

Perhaps the following resume 1 of my views on this point 
may be useful : — 

1. In obtaining (6) originally I employed the complete 
elastic solid equations for isotropic materials. In other cases 
where difficulties have arisen over expansion in series, they 
seem mainly due to the fact that the elastic solid equations have 
been whittled down in the first instance for purposes of 

338 JDr. C. Ohree on Longitudinal 

simplicity. When one omits terms in a differential equation for 
diplomatic reasons, the results may be perfectly satisfactory 
under certain limitations. Owing, however, to the mutilation 
of the differential equations, the resulting solutions are unlikely 
to contain within themselves any satisfactory indication of the 
limits to their usefulness. It is very much a case of running 
a steam-engine without a safety-valve. 

2. When the bar is of circular section and isotropic, the 
series occurring in (6) are Bessel's functions of a well-known 
type, whose rapidity of convergence appears well ascertained 
under the normal conditions of the problem. When the 
section is circular, and the material not isotropic but sym- 
metrical round the axis, the series, whose mathematical law of 
development I have obtained, converge to all appearance quite 
as rapidly as in the case of isotropy. The other cases of 
isotropic material — sections elliptical or rectangular — which 
I have considered present similar features ; the only difference 
being that the rate of convergence diminishes with increase 
in any one dimension of the cross section. 

3. If we suppose k = Q, or the vibrations to be of infinite 
period, the solution must reduce to that for the equilibrium of 
a rod under uniform longitudinal traction. ISTow, in the case 
of equilibrium 7/0 reduces to a constant, while a and ft are 
linear in x and y for all forms of cross section. The commencing 
terms in series (6) are thus of the proper form under all 
conditions, and the form of the differential equations shows 
that if a, for instance, contains a term in 00 it must contain 
terms in a? 8 , xy 2 , and other integral powers of % and y. 

4. The general type of the differential equations is the 
same for all kinds of elastic material, isotropic or ?eolotropic, 
and the surface conditions are identical in all cases ; thus the 
type of solution must always be the same. The results may 
become enormously lengthy for complicated kinds of seolo- 
tropy, but by putting a variety of the elastic constants equal 
to one another we must reduce the most complicated of these 
expressions to coincidence with the corresponding results for 
isotropy. Of course it does not follow that the convergence 
will be equally rapid for all materials. A large value of a 
Poisson's ratio in conjunction with an elongated dimension 
of the cross section may reduce the convergency so much as 
to throw the higher " harmonics " outside the pale of longi- 
tudinal vibrations. 

5. The more the section departs from the circular form 
the less rapid in general is the convergence, and the larger 
the correction supplied by the second approximation. In 
fact the size of the correction is probably the best criterion 

Vibrations in Solid and Hollow Cylinders, 339 

by which to judge of the limitations of our results. If the 
correction is large even for the fundamental note, it is pretty 
safe to conclude that the section is not one adapted for the 
ordinary type of longitudinal vibrations. If a section, for 
instance, were of an acutely stellate character, with a lot of 
rays absent and the centroid external to the material, I for 
one should be extremely chary of applying to it the ordinary 

§ 7. For deiiniteness let us consider the fixed-free vibra- 
tions, taking the origin of coordinates at the centroid of the 
fixed end. Our terminal conditions are 

7 = when z = 0, 
S=0 „ z = l; 
the latter condition being the same thing as 

_¥ =0 when z = L 

These conditions give at once 

6=0, p=(2* + l)w/2Z, 

and hence 

I pz sin {pz — e) dz = 1 cos (pz — e) dz. 

J Jo 

Take the axes of x and y along the principal axes of the 
cross section cr, so that 

\\ xy dx dy = 0, 

ij x 2 dx dy = k. 2 2 (t, Jj y 2 dx dy = k?<t. 

Then, substituting from (6) in (5), we obtain at once 

(Ep-Fp/ / >)(C + Crf + C 2 / V + ...) 

= -^(A 1 « 2 HB 1 V+...) . • (?) 

The section is supposed small, L e. terms ^ in k{ 2 , k 2 2 are 
small compared to O , though large compared to the terms 
of orders /q 4 , &c, which are omitted in the above equation. 
Thus as a first approximation the coefficient of C must vanish, 

*=?VE/p, (1) 

which is simply the ordinary frequency equation. 

■* Treating the other two equations of the type (5) similarly, 

340 Dr. 0. Cbree on Longitudinal 

we obtain the two results 

E(A 1 +3A 3 *^+A 3 'V + ...) 

= - W 9 1 3 (C + C 2 * 2 2 + CV V + ...)+ 9B1V - -W + . . . | , (8) 

E(B 1 ' + B 3 V + 3B 3 'V+...) 
_ -k 9 P ^ (C + (W + C 2 % 2 + ...) + V Ai/c 2 2 - B x V + . . . I . (9) 

The terms not shown are of the order /q 4 , k 2 4 , or higher 
powers of tc x and # 2 . Combining (8) and (9) we get 

E(A 1 -B 1 0=^p(l+7 ? )(A 1 ^ 2 -B 1 V)+..., . . (10) 

E (A, + B/) = - 2k* P ( v >p) (C + C 2 * 2 2 + Co V + . . .) 

+ (1-9,)^(A 1 / C2 2 +B 1 V) + ... • (11) 

To see the significance of (10) replace k^p by its approximate 
value /> 2 E , when we have 

A 1 -B/=(1 + ,){A 1 0« 2 )*-B 1 >* 1 )*}+.. • (12) 

As we have seen, p=(i + J)w/Z; and thus, so long as i is not 
too large, {pfc 2 )- and (/^J 2 are in a thin rod small quantities 
of the orders {kJI) 2 and (/cjl) 2 . Hence we deduce from (12) 
as a first approximation 

B/ = A ; (13) 

This is all we require for our present purpose ; but, in pass- 
ing, it may be noted that as a second approximation we have 

B^Mi+a+^Vi 2 -*/)}. 

The more the section departs from circularity — i. e. the 
more elongated it is in one direction — the greater is the dif- 
ference between B/ and A x . This and the fact that ik x jI and 
IkJI must both remain small are useful indications of the 
limitations implied in the method of solution. 

Employing (13) in (7) and (11) we have, neglecting 
smaller terms, 

(Bp-Afy&O (Co + CW + CoV) = -PprjA^ + K*), 
2k*p Wp) (C + C 2 * 2 2 + C 2 V) = - 2E Aj. 

Whence we deduce at once, without knowing anything of 
the constants C 2 , A 3 , &c, 

Wp-k* P /p)+(2P(»,/p) =t?M<h* + *i*)/(2E). 

Vibrations in Solid and Hollow Cylinders. 341 

Using in the small term (that containing & 4 ) the first ap- 
proximation (1), we have 

F /3 =E^{i-^ 2 (« l 2 +«/)}; • • • (14) 

and this, as 

simply reduces to (3). 

That the above proof is as satisfactory in every way as one 
based on ordinary elastic solid methods I should hesitate to 
maintain. Unless one knew beforehand a good deal about 
the problem there would, I fear, be considerable risk of mis- 

§ 8. In illustrating the method in detail I have selected 
the case of isotropic material simply because I did not wish 
to frighten my readers. The assumption of isotropy almost 
invariably shortens the mathematical expressions, and gene- 
rally also simplifies the character of the mathematical opera- 
tions ; and isotropic solids thus flourish in the text-books to a 
much greater degree than they do in nature. When, how- 
ever, the mathematical difficulties are trifling, as in the 
present case, it seems worth while considering some less 
specialized material. I shall thus briefly indicate the appli- 
cation of the method to the case of material symmetrical with 
respect to the three rectangular planes of x, y, z, taken as in 
the previous example. In this case the stress-strain relations 
involve, on the usual hypothesis, nine elastic constants. 
Such quantities as Young's modulus or Poisson's ratio must 
be defined by reference to directions. Thus let E x , E 2 , E 3 
denote the three principal Young's moduli, the directions I, 2, 3 
being taken along the axes of x, y, z respectively. There 
are six corresponding Poisson's ratios, each being defined by 
two suffixes, the first indicating the direction of the longitu- 
dinal pull, the second that of the contraction. For instance, 
rj 12 applies in the case of the contraction parallel to the 
y-axis due to pull parallel to the #-axis. The order of the 
suffixes is not immaterial, but there exist the following 
relations : — 

WEi=W E »; WEj=WEi.J W E 3 = W*V (15) 
The three equations answering to (4) are 

vjj fo dz dy dz= k * p Jj] frs-to^-^y}** *9 d ~, ( 16) 

Ei n)s^^^ = ^]Ij ^ ax ~ ri ^~" ni ^ dxd y dz ^ i 17 ) 

342 Dr. 0. Chree on Longitudinal 

From the nature of the elastic solid equations the expres- 
sions for the displacements must be of the same general form 
as for isotropy, so that we may still apply the formulae (6) 
for a, ft, y. Doing so, and following exactly the same pro- 
cedure as in the case of isotropy, we obtain from (16), for 
any shape of section, 

(E 3 p - k*plp) (C + ...) = - *V(%i A^ 2 + *AV) + • . -, (19) 

and from (17) and (18) as first approximations 

A 1 /(%3E 3 /E 1 )=B 1 '/( %3 E 3 /E 2 ) = -p(C +...). . (20) 

Thence we obtain at once 

k*p=r>E 3 {l-p*JZ 3 (^-W+ ?gV)} • (21) 
Employing (15) we give this the more elegant form 

*V -j*E»{ l -f ( Vsi W + % 2 V) \ , 


k=p(K 3 / P )Hl-i P *(vnW + V S 2W)\. ■ (22) 
For a circular section of radius a 

k=p(E i /p)i{l-±p*a*( V51 * + V32 *)}. . . (23) 

For a rectangular section 2a x 2b ; the side 2a being parallel 
to the #-axis, 

k=p(E 3 /p)h{l-ip*( a * V3] ? + b\ 3 /)\. . . (24) 

For a given size and shape of rectangle, the correction to 
the first approximation is largest when the longer side is that 
answering to the larger Poisson's ratio (for traction along 
the rod). Possibly experimental use might be made of this 
result in examining materials for seolotropy. 

If the material, though not isotropic, be symmetrical in 
structure round lines parallel to the length of the rod, 

V3 i =Vs2 = V, say, 
and writing E for E 3 in (22) we reproduce the result (60) of 
paper (C). 

The results (22), (23), and (24), so far as my knowledge 
goes, are absolutely new. 

Extension of Earlier Results. 

§ 9. My paper (A), like the corresponding investigation of 
Pochhammer, dealt only with a solid circular cylinder ; but 
the same method is applicable to a hollow circular cylinder. 
For greater continuity I shall employ in the remainder of 
tbis paper the notation of paper (A) . 

Vibrations in Solid and Hollow Cylinders. 343 

The displacements are u outwards along r, the perpendi- 
cular on the axis, and w parallel to the axis, taken as that of 
z. Thomson and Tait's notation 711, n for the elastic constants 
is employed. TJie frequency is k/2w and the density p, as in 
the earlier part of this paper, and for brevity 

&p/(m + n)=J, k 9 p/n = P, . . . (25) 

so that a. and ft have utterly different significations from 
their previous ones. 

There being no displacement perpendicular to r, in a trans- 
verse section, the dilatation 8 is given by 

5. du u dio , x 

S= A- + r + dz ^ 

It was shown in paper (A) that the following equations held 
#8 .ldB. d*8 

d? + v<h- + d;* +aB=0 > • • • • ( 2? ) 

*£ + !£_« +£+,<*,= _*<» (28 ) 
dr* r dr r 2 dz l n dr K ' 

d 2 w , 1 div , d 2 w -g m d8 ,_. 

f/r* 2 ?' dr a* 2 rc dz 

Employing J and Y to represent the two solutions of 
BessePs equation we find, as in paper (A), that the above 
equations are satisfied by 

8 = cos kt cos (pz - e) { G J {r (a 2 - f)h) + 0' Y (K" 2 -p 2 )h) } , (30) 
11= cos ft cos {pz-e) [AJ\{r(/3 2 -p 2 )i) + A'Y l (r{/3 2 -p^) 

~f i iE^\ GJ M^P 2 ) i ) +O r Y 1 (r(««-^*)}], (31) 
mj=- cos fa sin Cp*-e) R AJ (?'(/3 2 -^ 2 )*) + A'Y (r(/3 2 -p 2 )t) j- 
x (fag + ^j*^ {OJo(Ka»-^)») + PYo(r(a»-^)i)}] > (32) 

where A, A', C, C are arbitrary constants to be determined 
by the surface conditions. 

In reality ofi-p 2 is negative ; but the properties of the 
J and Y Bessel functions which at present concern us 
are not affected thereby. ft 2 —p 2 , on the other hand, is 

344 Dr. C. Chree on Longitudinal 

§ 10. If a and b are the radii of the outer and inner cylin- 
drical surfaces respectively, then from the conditions which 
hold over these surfaces we must have 

*+%-*> ™ 

(m-n)B + 2np=0, .... (34) 

when r = a, and when r=b. 

As regards the terminal conditions we should have, follow- 
ing the ordinary view of longitudinal vibrations, 

M? = over a fixed end, .... (35) 
a free end. (36) 

zz = (m — n)$-\-2n—=0, 

> over a 

^(S+f)=° * 

We have no means of satisfying these terminal equations 
by means of the present solution save by selecting suitable 
values for p and e. Clearly if both ends e = and z = l be 
fixed we accomplish our object by putting 

e = 0, p = iirjl. 

If, however, z = l be a free end, while ^ = is a fixed, we 
must have e = to satisfy the conditions of the fixed end ; 
and this leaves us with 

zz cc cos pi, 
zr oc sin/>Z 

over the free end. This is the difficulty we have already 
indicated in § 4 ; and it is in no respect peculiar to hollow 
cylinders, and need not further concern us at present. 

§11. In dealing with the surface conditions, brevity is 
effected by the use of the notation 

\=>v/a 2 -p 2 , /xeee V73 2 -;? 2 , . . (37) 

a 2 _/3 2 = \ 2 -Ar. 

After simplifications, into which I need not enter, the elimi- 
nation of A, A', C, C from the four equations holding over 
the cylindrical surface supplies the determinantal equation 

Vibrations in Solid and Hollow Cylinders. 



2p 2 A 


2p 2 X 

a (^ — i> ) 

m — n \ 2 T , 


a 2 (^ 2 -l' 2 ) 
2x> 2 X 

%_ X 2 

m— n 


h — w. X 2 

2n/u oV J a 2 // 

This equation is true irrespective of the relative magnitudes 
of a and b. It constitutes a frequency equation supplying 
values of k which apply to the type of vibrations consistent 
with the surface conditions. If both ends of the rod be fixed 
there is no restriction to the absolute values of a/l and b/l; 
but if one or both ends are free, such a restriction is really 
involved in the fact that unless ia/l and ib/l be both small — 
i being the order of the harmonic of the fundamental note 
under consideration— the failure to satisfy exactly the terminal 
condition ^ 

zr — 

involves an inconsistency which cannot be allowed. 

§ 12. The case of a thick rod fixed at both ends is of little 
physical interest, and the treatment of (38) in its utmost 
generality would involve grave mathematical difficulties. 
I thus limit my attention to the case when ia/l and ib/l are 
both small. This implies that a\, b\ ayb i and b/n are all small. 
Thus in dealing with the various Bessel functions we may use 
the following approximations*, which hold so long as the 
variable x is small, 

J (#) = 1-^/4, 

^(^ = 1^(1 -<z> 2 /8), 

Y (*) = (1-^/4) log tf + 074, ^ ' ( 39 ) 



YiW = |(l- 

x 2 / '8) log x— # _1 — w/4tj 

Y/(a ? )=i(l-|^)log^ + ^- 2 + |. J 

Ketaining only the principal terms in (38), we are of course 
led at once to the first approximation (1). Again, if (1) held 
exactly w T e should find 

« 2 =p 2 n (3m — n) -f- m (m + n) , 

fju 2 — p 2 =p 2 (m — n) /m ; 
* Cf. Gray and Mathews' ' Treatise on Bessel Functions,' pp. 11, 22, &c. 


346 Dr. C. Chr'ee on Longitudinal 

and to a first approximation 

J\ (aX) 2p 2 X 2p 2 X 2 _2 m(m-n) 

Ji(ajj,) a\fjb 2 —p 2 ) <z 2 fi(fjb 2 — p 2 ) fi n{3m—n) 9 

m—n J (aX) X 2 J/(aX) _ 2(m— n) X 2 _ 2m(m — n) 

2nfjb Ji(a/ji) a 2 /n J / (aft) 2n/j, afi fin^dm — nf 

and similarly if a be replaced by b. 

We thus see that the third column in the determinant (38) 
is such that each principal term in it is obtained by multi- 
plying the principal term in the same row in the first column by 
the same constant 2m (m ~ n)-h- \/jbn(3m — n) \. Now if one 
column of a determinant is obtainable by multiplying another 
column by a constant that determinant vanishes. It is thus 
at once clear that in proceeding even to a second approxima- 
tion we need retain only the principal terms in the second 
and fourth columns of (38). This removes what seemed at 
first sight a formidable obstacle, viz. the occurrence of the 
logs in the expressions for Y , Y x , and Y/. 

§ 13. For further simplification of the determinant multiply 
each term by 2, and divide the first and second rows by a/n 
and bfi respectively. Then for the second row write the 
difference between the first and second rows, and for the 
fourth row the difference between the third and fourth rows, 
and multiply the resulting rows by b 2 /(a 2 — b 2 ). Finally 
multiply the second column by a 2 fi 2 /2, the third column by 
[Aa?, and the fourth column by a 2 fji<z 2 /2. We thus reduce (38) 
to the easily manageable form 

i-ia,y -l _^>! 2 (i_i^) -M, 

[M Z —p~ K ° ' fl 2 —p 2 

xyb 2 2 P 2 

1 A2..2 1 

8 T H^-p 2 ) fi 2 -p 2 

1-faV 1 x» + ^^-^(^ + W , ?=5) 1 

%hY 1 ib 2 X 2 (3X 2 + 2a 2 ^) 1 

After algebraic reduction, use being made in the secondary 
terms of the first approximation results (1), (40), &c, we 
easily deduce from (41) 




»-**^{i-v(*£](!*+»>}.- («> 

l=p(E/p)i{l^^rf a -±ty . . . (43) 

Vibrations in Solid and Hollcw Cylinders. 347 

In a hollow circular section the radius of gyration round 
the perpendicular to the plane through the centre is given by 

K 2 =(a 2 + b*)/2, 

so that (43) is in agreement with (3) and (22). 

The fact that (43) is merely a special case of (3) or (22) 
may seem to indicate that our separate treatment of the 
hollow cylinder, or tube, is quite unnecessary. I can only 
say that having regard to the methods by which (3) and (22) 
were arrived at — more especially to the fact that in establishing 
(6) I was dealing with solid cylinders — I had long felt the 
desirability of an independent investigation. 

§ 14. The complete determination of the constants A, A 7 , 
C, C^, and of the several displacements, strains, and stresses 
to the degree of accuracy assumed in (43), though not a very 
arduous labour, would require more time than seems warranted 
by the physical interest of the problem. I thus confine my 
further remarks to the form of the longitudinal displacement 
w. Substituting their approximate values for tbe J's and 
Y's from (39) in (32), we find 

piDJfJb cos M sin (pz — e) = 

- A(l - i/zV) ~ A'{ (1 - i|*V) log fir + JpV \ 

Now, considering only their principal terms, it is easily 
seen that A'/ A and C/C are both of the order (p 2 ab) 2 . Thus, 
to the present degree of approximation, we may leave the A' 
and G / terms in iv out of account. Also confining ourselves 
to principal terms, we easily find 

-A = Cp 2 /fJLCC 2 

2(m—n) ~ m + n 
Hence, employing the two last of equations (40), we deduce 
i{ Afi 2 - (Cp 2 /y.a 2 )\ 2 \ H- { - A + Cp 2 /f*« 2 } 

We thus have from (44), to the degree of approximation 
reached in (43), 

iv = iv cos kt sin (pz — e) (1 — i^pV) , . . (45) 

where iv is a constant which depends on the amplitude of the 

The expression (45) for iv is exactly the same as I found 

348 Longitudinal Vibrations in Solid and Hollow Cylinders. 

in my earlier papers for solid cylinders ; the r 2 term repre- 
senting one of the additions I deem it necessary to make to 
Lord Ray lei gh's assumed type of vibration. 

The paraboloidal form of cross section, met with except at 
nodes or when cos kt vanishes, seems to me an interesting 
feature of the longitudinal type of vibrations. Possibly, 
observations on light reflected from a polished terminal face 
might lead a skilled experimentalist to interesting conclusions 
as to the value of n. 

It should, however, be borne in mind that, inasmuch as the 

terminal condition zr = is not exactly satisfied by the above 
solution in fixed-free vibrations, there may be a slight departure 
from the theoretical form in the immediate neighbourhood of 
a free end. 

§15. The result (43) is true irrespective of the relative 
magnitudes of b and a. If h/a be very small, the correctional 
term is the same as for a solid cylinder of the same external 
radius. If, on the other hand, b/a be very nearly unity, or 
the cylinder take the form of a thin- walled tube, we have 

Jc=p(E/p)Hl-hP*vW) (46) 

The correctional term is here twice as great as in a solid 
cvlinder of radius a. 

* § 16. An experimental investigation into the influence of 
the shape and dimensions of the cross-section on the frequency 
of longitudinal vibrations is certainly desirable. In com- 
paring the results of such an investigation with the theoretical 
results here determined, several considerations must, however, 
be borne in mind. 

Statical and dynamical elastic moduli are to some extent 
different, so that the value of E occurring in (2) or (3) is not 
that directly measured by statical experiments*. In other 
words the difference between the observed frequency of the 
fundamental vibration in a fixed-free bar, and the frequency 
calculated from the ordinary formula 

if E be determined directly by statical experiments, is not to 
be wholly attributed to the defect of the ordinary first 
approximation formula. Again, it must be remembered that 
E varies t, often to a very considerable extent, in material 
nominally the same ; so that the difference of pitch observed 

* See Lord Kelvin's Encyclopaedia article on Elasticity, § 75, or 
Todhunter and Pearson's ' History,' vol. iii. art. 1751. 

t For the effects of possible variation in the material throughout the 
bar, see the Phil. Mag. for Feb. 1886, pp. 81-100. 

Experiments on Artificial Mirages and Tornadoes. 349 

in different rods cannot without further investigation be safely 
ascribed to differences in the area or shape of their cross- 
sections. Further, elastic moduli may alter under mechanical 
treatment, so that it would be unsafe to assume if a hollow 
bnr were further hollowed or were altered in shape, that its 
Young's modulus would remain unaffected. 

If it were possible to measure with sufficient accuracy the 
frequency of the fundamental note and several of its 
" harmonics " in a single rod, one would have a more certain 
basis of comparison with the theoretical results. Even in 
this case, however, there is the consideration that in practice 
the rod must be supported in some way, and this is likely to 
introduce some constraint not accurately represented by the 
theoretical conditions. Again, reaction between the vibrating 
rod and the surrounding medium may not be absolutely 
without influence on the pitch *. 

I mention these difficulties because their recognition may 
prevent a considerable waste of time on the part of anyone 
engaged in experiments on the subject. 

Though somewhat of a side issue, it may be worth remark- 
ing that the correction factor l — ^p^ific 2 for the frequency in 
isotropic material contains no elastic constant except Poisson's 
ratio. Thus observations made on rods differing only in 
material might throw some light on the historic question 
whether rj is or is not the same for all isotropic substances. 

The discussion of equations (6) and of the experimental 
side of the problem has been largely expanded at the suggestion 
of the Society's referee. 

XXX. Some Experiments on Artificial Mirages and 
Tornadoes. By R. W. Wood f. 
[Plate III.] 

IN an article published in i Nature ' for Nov. 19, 1874, 
Prof. Everett, in discussing the phenomenon of mirage, 
showed that the condition necessary for the formation of sharp 
images in a horizontally stratified atmosphere, is a plane of 
maximum refractive index, the optical density decreasing as 
we go above or below this plane in direct proportion to the 

A horizontal or nearly horizontal ray will be bent towards 
and cross the plane of maximum density, where it changes 
its curvature and is again bent towards the plane, which it 

* Cf. Lamb, Memoirs and Proceedings Manchester Phil. Society, 
vol. xlii. part iii. 1898. 

f Communicated by the Author. 

Phil. Mag. S. 5. Vol. 47. No. 287. April 1899. 2 B 

350 Mr. R. W. Wood on some Experiments 

may thus cross again and again, traversing a path which is 
approximately a sine curve. 

While showing the curve-trajectory of a ray of light in a 
vessel filled with brine, the density of which increased with 
the depth, it occurred to me that by properly regulating the 
refractive index of the liquid, the ray might be made to 
traverse a sine curve. 

Some attempts in this direction were so successful, and 
yielded such beautiful experiments for the lecture-room, that 
it seems worth while to publish them, together with some 
photographs of the trajectories, although, as I have since 
learned, very similar experiments have already been described 
by Wiener ( u Gekriimmte Lichtstrahlen," Wied. Annalen, 
xlix. p. 105). 

For the liquid 1 adopted an arrangement very similar to 
the one described by Prof. Everett for obtaining mirage in a 
rectangular tank. 

A trough was first made of plate glass, about 50 cms. long, 
10 cms. bigh, and 2 cms. wide. This was filled to the depth 
of 3 cms. with a concentrated solution of alum. By means of 
a pipette, of the form shown in PI. III. fig. 1, water containing 
about 10 per cent, of alcohol was carefully deposited on the 
alum solution to a depth of 3 cms. The addition of the alcohol 
brings up the refractive index of the water, and is necessary 
for reasons that will be spoken of presently. As a liquid of 
high index, with a specific gravity intermediate between that 
of the other two liquids, I used, instead of sugar and whiskey, 
a mixture of glycerine and 85 per cent, alcohol, the right pro- 
portions being easily found by experiment. 

The mixture should float on the alum solution and sink in 
the water, and is introduced between the two layers by means 
of the pipette at the end of the trough through which the ray 
is to enter. A layer about 3 cms. in thickness will be found 
about right. 

All three of the solutions should be first acidified with a few 
drops of sulphuric acid, and rendered fluorescent with quinine. 
The difference in surface-tension between the two upper 
layers may give some trouble: when the pipette is withdrawn 
it may draw a thread of the glycerine and alcohol mixture up 
through the water, and a complete upsetting of the layers 
occur as a result of the forces of surface-tension. 

This invariably happens when alcohol is not added to the 
water, and can be remedied either by the addition of water to 
the glycerine mixture or of more alcohol to the water. It is 
a good plan in withdrawing the pipette to pull it out slowly 
in a very oblique direction, in order that the heavy liquid may 
be washed off before the tube reaches the surface. 

on Artificial Mirages and Tornadoes. 351 

The three layers may now be cautiously stirred to hasten 
the diffusion, after which they should be allowed to rest a few 
minutes until the strias have disappeared. 

If now a beam of light from an arc-lamp, made parallel by 
means of a condensing-lens, be thrown obliquely into one end 
of the trough, it will be seen to traverse the liquid in the 
form of a most beautiful blue wave, the curvature of which 
varies with the angle at which the ray enters. Rays of 
light travelling in sine curves are shown in figs. 2 and 3, 
which were photographed directly from the trough. 

Prof. Everett showed in his paper that a parallel or slightly 
divergent ray entering a medium of this description would, 
converge to a linear focus, and then successively diverge 
and converge to conjugate foci. This phenomenon is shown 
in fig. 4, which is a photograph of the trough with a rather 
wide beam of horizontal parallel light entering the end. 
This experiment I have never seen described, though Exner 
has shown that the eyes of some insects operate in a similar 
manner, the visual organ consisting of a transparent cylin- 
drical body, the axis of which has a high refractive index, 
while as we approach the surface the optical density decreases 

The beautiful miniature desert-mirages that I have witnessed 
on the level city pavements of San Francisco (see letter and 
photograph in ' Nature ' for Oct. 20, 1898), suggested to me 
the idea of reproducing this phenomenon on a small scale in 
the class-room. 

Although 1 have already described very briefly an expe- 
riment of this nature, I will repeat the description now some- 
what more in detail. Three or four perfectly flat metal 
plates, each one about a metre long and 30 cms. wide, should 
be mounted end to end on iron tripods and accurately levelled. 
The plates should be thick enough not to buckle when heated, 
say 0'5 cm. I have used plaster plates, made by casting 
plaster of Paris on plate glass, with some success, though they 
are fragile and not very durable. 

Probably plates of slate would serve admirably, since they 
will stand a fair amount of heating, and can be obtained very 
flat and smooth. 

The plates must be thickly sprinkled with sand to destroy 
all traces of reflexion at grazing incidence, and the sand sur- 
face should appear perfectly level when looked along from a 
point just above its plane. On the absolute flatness of the 
desert depends the successful working of the experiment; 
therefore too much care cannot be given to the adjustment of 
the plates. An artificial sky must be formed at one end of the 
desert. If the experiment is being performed at night, a 

2 B2 

352 Experiments on Artificial Mirages and Tornadoes. 

sheet of thin writing-paper with an arc-lamp behind it works 
very well, but a large mirror set in a window and reflecting 
the sky is better, when daylight is to be had. 

Between the sky and the desert a small range of mountains, 
cut out of pasteboard, should be set up. The individual peaks 
should be from 1 to 2 cm. high, and the valleys between them 
should be only a trifle above the level of the desert. The general 
arrangement is shown in fig. 5. 

The plates are now heated by means of a row of burners, 
which should be moved about from time to time in order to 
prevent overheating any one place. 

If now we look along the desert, holding the eye only a 
trifle above the level of the sand, we shall see the mountains 
sharply outlined against the sky: as the temperature rises a 
lake begins to form in front of the mountain-chain, and in a 
few moments the inverted images of the peaks appear as if 
reflected in the water. If the eye be depressed a trifle, the 
base of the mountain-chain vanishes completely in the illusory 
lake, which now appears as an inundation. These appear- 
ances are shown in fig. 6, the photographs having been taken 
of the actual mirage on the artificial desert. The first of the 
three shows the appearance when the plates are cold, the 
second the apparent lake with the images of the peaks in the 
water, and the third the vanishing of the lower portions of 
the range. Two or three palm-trees, cut out of paper, were 
stuck up to add to the effect. Vertical magnification can also 
be shown on the hot desert: if the mountains are removed 
and a small marble be laid on the sand at the farther end of 
the desert, it will be found that if the eye be brought into 
the right position, the circular outline will change into an 
ellipse, and as the eye is lowered the image will contract to a 
point and eventually disappear. The magnification in this 
case is of course due to the running together of the direct 
and refracted images. I have observed similar cases in 
looking across our lake, when the water was warm and the 
air cold, patches of snow on the opposite shore, too small to 
be visible to an eye several metres above the level of the lake, 
coming out very distinctly when one walked down a bank to 
the water's edge. 

The atmospheric conditions existing when mirages of this 
description are observed are such as give rise to the dust- 
whirls, so often seen on the American desert, and when 
existing on a larger scale, to tornadoes. There seemed no 
reason why these whirlwinds should not be produced on a 
small scale as well as the mirages. One of the metal plates 
was sprinkled with precipitated silica and heated with a few 
burners : in a few minutes most beautiful little whirlwinds 

On the Thermal Properties of Normal Pentane. 353 

began to run about over the surface, spinning the fine powder 
up in funnel-shaped vortices, which lasted sometimes ten or 
fifteen seconds. The silica powder must be made by igniting 
the gelatinous precipitate formed when silicon tetrafluoride is 
conducted into water. The commercial article is not suffi- 
ciently light and mobile. 

Whirls formed in this way cannot be seen by a large 
audience, however, and I accordingly sought some way of 
making them on a larger scale. The plate was well heated 
after removing the silica and then dusted with sal-ammoniac: 
dense clouds of white vapour immediately arose from the hot 
surface, and presently in the centre there mounted to a height 
of about 2 metres a most perfect miniature tornado of dense 
smoke. By placing the plate in the beam of a lantern in a 
dark room, the whirls can be shown to a class in a large 
lecture-room. I find that it is best to put on the sal-ammoniac 
first and then heat the plate: the vortices then come off the 
plate almost continuously, and often persist for some time. 

An instantaneous photograph of one of these tornadoes was 
taken in bright sunshine, and is reproduced in fig. 7. 

This method of showing atmospheric vortices seems far 
preferable to the old way of forming them, by means of a 
rapidly whirling drum with cross partitions, as the whirls are 
produced by the same causes and under the same conditions 
that they are in nature. 

Physical Laboratory 

of the University of "Wisconsin, 

Madison, Nov. 20. 

XXXI. On the Thermal Properties of Normal Pentane. By 
J. Kose-Innes, M.A., B.Sc, and Sydney Young, D.Sc, 

IX the year 1894 an experimental investigation of the 
relations between the temperatures, pressures, and 
volumes of Isopentane, through a very wide range of volume, 
was carried out by one of us, and the results were published 
in the Proc. Phys. Soc. xiii. pp. 602—657. It was there shown 
that the relation p = bT — a at constant volume (where a and 
b are constants depending on the nature of the substance and 
on the volume) holds good with at any rate but small error 
from the largest volume (4000 cub. cms. per gram) to the 
smallest (1*58 cub. cms. per gram). 

In the neighbourhood of the critical volume (4*266 cub. 
cms.), and at large and very small volumes, the observed 
deviations were well within the limits of experimental error 
* Communicated by the Physical Society: read December 9, 1898. 


Mr. J. Rose-Innes and Dr. S. Young on th 


but at intermediate volumes they were somewhat greater, and, 
as they exhibited considerable regularity, it is a question 
whether they could be attributed entirely to errors of experi- 
ment. In any case, the relation may be accepted as a close 
approximation to the truth. 

A quantity of pure normal pentane having been obtained 
by the fractional distillation of the light distillate from 
American petroleum, it was decided to carry out a similar 
investigation with this substance; but, as it had been found 
that isopentane vapour at the largest volumes behaves 
practically as a normal gas, it was not considered necessary 
to make the determinations through so wide a range of 

The method employed for the separation of the normal 
pentane from petroleum has been fully described in the Trans. 
Chem. Soc. 1897, Ixxi. p. 442 ; and the vapour-pressures, 
specific volumes as liquid and saturated vapour, and critical 
constants have been given in the same journal (p. 44G). 

The data for the isothermals of normal pentane were 
obtained by precisely the same experimental methods as in 
the case of isopentane, and reference need, therefore, only be 
made to the previous paper (loc. cit.) . 

There were four series of determinations ; and particulars as 
to the mass of pentane, and the data obtained in each series 
are given below : — 


Mass of 
[Xormal Pentane. 

Data obtained. 



i IV. 




Volumes of liquid to critical point ; volumes above 
critical temperature to 280°. 

Volumes of unsaturated vapour from 140° to critical 
point ; volumes above critical point to 280°. 

Volumes of vapour at and above 80°. 

Volumes of vapour at and above 40°. 

The correction for the vapour-pressure of mercury was 
made in the same way as with isopentane : when liquid was 
present it was assumed that the mercury vapour exerted no 
pressure ; in Series I., above the critical point, one-fourth 
of the maximum vapour-pressure of mercury was subtracted; 
in Series IT. one-half ; in Series III. three-fourths ; and in 
Series IY. the full pressure. 

The volumes of a gram of liquid and unsaturated vapour 
are given in the following table. 

Thermal Properties of Normal Pentane, 355 

Volumes of a Gram of Liquid and of Unsaturated Vapour. 



rp j Pressure. 
1 ' millim. 

Volume, rn i 
cub. cm. P" 





V lume. 


cub. cm. 


c u . cm 

130°. 7023* 

2-022 170°. 







2-018 (cont) 










































; 35560 








































2 085 
















4 106 


2 056 




























































































































2 249 











































































































Pressure below vapour-pressure. 

356 Mr. J. Rose-Innes and Dr. S. Young on the 
Series I. (continued). 











cub. cm. 


cub. cm. 


cub. cm. 





















































! 280°. 












6 061 


; 4-498 
































' Volume. 



cub. cm. 


cub. cm. 


cub. cm. 





























19-56 i 


26 93 







1 22300 




16-32 I 




| 23130 



14 95 










7 61 












6 23 















9-45 i 






853 I 





























1631 1 










17 68 

















27 01 
























19 57 



























23 24 






























Thermal Properties of Normal Pentane. 
Series IT. (continued). 


„ Pressure, 








Tem P- millim. 

cub. cm 


cub. cm. 


cub. cm. 

260°. 33970 








(cont.) 36340 





(cont) i 32550 

10 37 














5 78 











6 70 













280°. i 13980 






: 15230 







Series III. 

T I 






r n Pressure. 




cub. cm. 



cub. cm. 



cub. cm . 

80°. ' 











(cont.)' 13930 






240°. 3631 



10529 : 




















































































44 07 
























29 76 



















280° 3933 







105 83 












































58 60 




































37 04 















47 63 


















| 36-90 






358 Mr. J. Kose-Innes and Dr. S. Young on the 

Series IV. 












cub. cm. 


cub. cm. 


cub. cm. 
















151-5 ! 

I 60 f '. 


322 6 










































94-6 i 







358-7 i 


















289 3 , 







266-1 ! 























































128 8 











































3360 | 







312-8 1 










128 5 
















242 3 
















174-6 ! 







163-2 ! 


































219 9 




















94-3 | 


Thermal Properties of Normal Pentane. 359 

Relation of Pressure to Temperature at Constant Volume. 

For the smaller volumes isobars were first constructed 
from the isothermals, and the temperatures at definite volumes 
were read from the isobars. The data from which the 
isobars were constructed are given below : — 

Isobars read from Isothermals. 


' 130°. 













Volume in cub. cms. 




























2 0254 










2 0790 















































2' 61 80 






, 2-0816 






2 7330 

; 2 9590 







| 2-8460 

111 the following tables the data for the isochors are given ; 
those for small volumes were read from the isobars, and those 
for larger volumes from the isotherms. 

Isochors read from Isobars. 

































































































199 05 






































Mr. J. Rose- limes and Dr. 8. Young on the 


read from 








38. i 4-0. 


4-3. 4-6. 






























i 220 




















































52500 , 

Isochors read from 



5 - 5. 

6. I 6-5. 





10. ! 12. 







1 " 


















i 28940 



! 31920 











1 34800 



! 37790 




3 750 


281 ib 

2 150 




! 40740 



! 43590 




3 300 







1 46400 













Isochors read from Isothermals. 


18. j 20. 
















.. ... 






































































21465 19690 










Thermal Properties of Normal Pentane. 
Isochors read from Isothermals. 
















o . 

1 40 


i 60 



























































































The values of b and a in the equation p = bT — a were 
obtained graphically from the preceding data. As with iso- 
pentane, the deviations are exceedingly small at the largest 
and smallest volumes and about the critical volume, but are 
larger at intermediate volumes ; they exhibit a similar regu- 
larity and are in the same direction as with isopentane. Here 
again the relation p = bT — a at constant volume, if not abso- 
lutely true, may be taken as a very close approximation to the 

In studying the variation of b and a with the volume it 

w r as found convenient, in the case of isopentane, to plot the 

. , 10,000 . e 10 10 . , i , ... , . . 
values or — ] and or — T against v 3 : and this has also been 

V ^r, _! 10* 

done for normal pentane. The values of b } a, v 3 ; -y— , and 

10 10 . 

for a series of volumes are given in the table below and, 



for the sake of comparison, the corresponding values of 

10 io bv 

— - for isopentane are added 

values of 10 10 /av 2 

(Table p. 362.) 
plotted against v 

in the 



The values ol . 
diagram on p. 363. 

In a former paper by one of the authors (Phil. Mao-, xliv. 
p. 77) it was pointed out that, besides the quantities b and a, 
it is often useful to consider a fresh quantity r, which is 
defined as follows : — For each volume there is one and onlv 
one temperature at which the gas has its pressure equal 
to that given by the laws of a perfect gas : this temperature 
is denoted by t. It is also shown that the numerical value 

362 Mr. J. Rose-lnnes and Dr. S. Young on the 





10 10 

jav 9 : 

in c.c.'s. 



.From drawn isockors. 















811,210 - 


















3 028 
























































































































6 040 



































2816 ; 






























































































































1957 i 


















































5 030 














































1770 j 

Thermal Properties of Normal Pentane. 


G O 

D ^„ 

c ( 




G ° 




08 c 












of t is given by the expression 

; making use of the 


values of a and b already given for normal pentane, the values 
of t have been calculated, and the results are given in the 
folio win £ Table : — 











833 9 








































848 9 










































845 8 















364 Mr. J. Kose-Innes and Dr. S. Young on the 

An examination of this table shows that r remains fairly 
constant for all large volumes down to about vol. 8. The 
actual numbers obtained vary a good deal ; but these variations 
are sometimes in one direction and sometimes in another, and 
there is no steady increase or decrease. It appears, then, 
that all the values of r above vol. 8 could be treated as the 
same without introducing any serious error ; this occurred 
likewise in the case of isopentane. What is still more note- 
worthy is that the same constant value of r could be used for 
both normal pentane and isopentane, keeping within the 
limits of experimental error. The mean value of t for all 
volumes above 8 was found to be 842*4 for isopentane ; it is 
838*5 for normal pentane ; and the intermediate value 840 
could be used in both cases without introducing any error 
greater than the unavoidable errors of experiment. 

When we pass on to the neighbourhood of the critical 
point, the value of r diminishes steadily as the volume 
decreases. For the critical volume itself t is about 807, and 
for vol. 2 it has sunk to 488. 

The most important conclusion arrived at in the case of 
isopentane was that the molecular pressure a does not follow 
a continuous law, but passes abruptly from one law to 
another somewhere about vol. 3*4 (Phil. Mag. xliv. p. 79). 
This inference was based on the study of a diagram in which 

the quantity — ^ was plotted against t>~*, and there appeared 

to be considerable evidence of discontinuity in the neighbour- 
hood of the volume already mentioned. Of course it is 
impossible to prove discontinuity of slope by means of a 
series of isolated points, but it is suggested very strongly ; 
and even if there be not discontinuity in the true mathematical 
sense of the term, there seems to be such a rapid change of 
behaviour as to amount practically to the same thing. 

It was therefore a matter of some interest to discover 

whether the diagram obtained by plottino- — - against v~i in 

the case of normal pentane would exhibit the same peculiarity. 
The diagram is given on p. 363, and it is easily seen that we 
have here a similar suggestion of discontinuity in the slope of 

— 2* this occurs somewhere about vol. 3*4, as with isopentane. 

In attempting to find a formula for the pressure of normal 
pentane we are therefore confronted with the possibility that 
we may require two distinct algebraic equations. We may 
simplify the problem considerably by confining our attention 

Thermal Properties of Normal Pentane. 


to volumes lying above 3*4 ; and this limitation still leaves us 
with all those conditions of the substance in which we can 
most usefully compare it with isopentane. 

Looking at the table on p. 362, which gives the series of 

values of — # an( * comparing it with the similar table for 

isopentane (Proc. Phys. Soc. xiii. pp. 654, 655), we notice 

that at the same volume the value of — t, is always smaller in 

the former case than in the latter. The difference is not great, 
but it remains too persistently with the same sign for us to 
disregard it. As we proceed to larger and larger volumes, 
however, the difference diminishes on the whole, and an 
interesting question arises whether we should be justified in 
treating it as ultimately vanishing when v is made infinite. 






o © 










> n 








D © 

° hi 









To elucidate this point a diagram was drawn in which the 

differences of — ^ between isopentane and normal pentane 

were plotted against v~* ; this diagram is reproduced 
above. The diminution in the differences with increase of 
volume is well shown in spite of the " wobbling " at large 
Phil. Mag. S. 5. Vol. 47. No. 287. April 1899. 2 G 

366 Mr. J. Rose-Tnnes and Dr. S. Young on the 


volumes; and a line running through the points might 
apparently end at zero difference. But though this result 
might be accepted as consistent with the experimental 
evidence if there were independent grounds for believing in 
it, it cannot be considered as the most probable judging 
solely from the diagram ; we should be rather led to believe 

that even at infinite volumes the value of — : . for isopentane 

remained larger than that for normal pentane. 

The above results respecting a and t are chiefly interesting 
because they seem capable of throwing some light on the 
vexed question of the influence exerted by difference of 
chemical structure on the thermal properties of a substance. 
Concerning this matter very little is known at present ; but 
it is common knowledge among organic chemists that two 
substances may have the same chemical composition and 
show practically the same behaviour whilst in the condition 
of rare vapour, and yet they may differ considerably as to 
their thermal properties in the liquid state. The great field 
of observation in which the substances lie between the con- 
ditions of a rare vapour and a common liquid has been left 
almost entirely unexplored. This gap in our knowledge 
makes it impossible to say in what precise manner the differ- 
ence between two isomeric substances originates ; whether it 
arises conjointly with the first deviations from Boyle's law, 
or whether the difference remains inappreciable even with 
increasing density until we reach the neighbourhood of the 
critical point. We may put the problem more precisely as 
follows : — If we imagine the pressure given by a series of 
ascending powers of the density, what is the lowest power of 
the density which has different coefficients for two isomeric 
substances ? 

We are now able to answer this question with a fair amount 
of exactness in the case of the two isomers, normal pentane and 
isopentane. If, as seems most probable, there is a difference 

between the — ^ for normal pentane and the — T for iso- 

pentane, even at infinitely large volumes, this shows that the 
coefficients of the second power of the density in the expan- 
sion of p must be different for the two substances. On the 

other hand, if there is no difference between the — « for 

-j avr 

normal pentane and the — ^ for isopentane at infinitely large 

volumes, then the coefficients of the second power of the 
density in the expansion for p must be the same, since t has 

Thermal Properties of Normal Pentane. 367 

already been shown to have the same value for the two sub- 
stances at infinite volumes ; and the lowest power of the 
density which has different coefficients for the two isomers 
must be the third. 

It w r as thought advisable to test these conclusions by a 
different method. In a former paper by one of the authors 
(Phil. Mag. xliv. p. 80 ; see also Phil. Mag. xlv. p. 105) it 
was shown that in the case of isopentane we might reproduce 
the original observations very closely by putting 

* v \ v + k — gv~ 2 ) 

gv~ 2 ) v(v + k) 9 

where R, <?, k, g, and I are constants. If we assume that this 
formula holds also for normal pentane, and if it be true that 
the difference of pressure between normal pentane and iso- 
pentane at the same temperature and volume varies ultimately 
as the third powder of the density, then we should be able to 
reproduce the experimental data for normal pentane by 
means of the above formula, giving to R, e, and I the values 
already found for isopentane. We may accordingly take 
11 = 863-56, £ = 7-473, 1= 5420800, and we still have the 
constants k srndg left at our disposal to meet the requirements 
of the normal pentane data. On examining the observations 
given in Series I. of this paper we find that we can con- 
veniently put k — 3*135, #=6'695, and we have to test how 
far the formula with these constants reproduces the experi- 
mental results given in Series II., III., and IV. In order to 
institute an effective comparison between theory and observa- 
tion a diagram was made in which pv was plotted against v~i ; 
the calculated isothermals were drawn as continuous lines, 
while the experimental values were put in as dots. An 
examination of the diagram shows that a fair concordance 
between calculation and experiment has been secured ; but 
the agreement is not so good as could be wished. Deviations 
amounting to 1 per cent, are not uncommon, and in places 
they approach 2 per cent. If we have regard to the differ- 
ences which often occur in inquiries of this kind between the 
results of independent observers, we might conclude that the 
above deviations are unimportant, and that R, e, and I were 
really the same for the two pentanes as supposed. But it 
seems more likely that the deviations are too large to be 
neglected ; hence, the most probable inference is that the I 
for normal pentane is not the same as the I for isopentane, 
thereby confirming our former conclusion. 

2 C2 

[ 368 ] 

XXXII. An Application of the Diffraction- Grating to Colour- 
Photography. By B. W. Wood*. 

IF a diffraction-grating of moderate dispersion and a lens 
be placed in the path of a beam of light coming from a 
linear source, and the eye be placed in any one of the spectra 
formed to the right and left of the central image, the entire 
surface of the grating will appear illuminated with light of a 
colour depending on the part of the spectrum in which the 
eye is placed. If one part of the grating has a different 
spacing from the rest, the spectrum formed by this part will 
be displaced relatively to the first; and if the eye be placed 
in the overlapping part of the two spectra, the corresponding- 
portions of the grating will appear illuminated in different 
colours. This principle I have made use of in the develop- 
ment of a new method for producing photographs in natural 
colour. I have eliminated the use of pigments and coloured 
screens entirely in the finished picture, the photograph being 
nothing more nor less than a diffraction-grating of variable 
spacing, the width between the lines in the different parts of 
the picture being such as to cause them to appear illuminated 
in their proper colours when view r ed in the manner described. 

We will take at the start three diffraction-gratings of such 
spacing that the deviation of the red of the first is the same 
as that of the green of the second and the blue of the third 
(the red, green, and blue in question being of the tints of the 
primary colours of the Young-Helmholtz theory of colour- 
vision). If these three gratings be mounted side by side in 
front of a lens their spectra will overlap; and an eye placed 
in the proper position will see the first grating red, the second 
green, and the third blue. If the first and second be made 
to overlap, this portion will send both red and green light to 
the eye, and will in consequence appear yellow. If all three 
be made to overlap in any place, this place will send red, 
green, and blue light to the eye, and will appear white. 

The method that I first employed to produce photographs 
showing natural colours on this principle is the following 5— 
Three negatives were taken through red, green, and Mue 
screens in the usual manner : from these, positives were made 
on ordinary lantern-slides (albumen-slides are necessary for 
reasons which I will speak of presently) . The positives, when 
dry, were flowed with bichromated gelatine and dried in 
subdued light. The three diffraction-gratings of proper 
spacing, ruled or photographed on glass, w^ere placed over 
these positives, and exposed to the sun or electric light for 
* Communicated bv the Author. 

Diffraction- Grating and Colour-Photography . 369 

thirty seconds. On washing these plates in warm water, 
diffraction-gratings of great brilliancy were formed directly 
on the surface of the film. Albumen-plates must be used, 
since the warm water softens and dissolves a gelatine film. 
Three sheets of thin glass, sensitized with the bichromated 
gelatine, were placed under the three positives, and prints 
taken from them. The portions of each plate on which the 
light had acted bore the impression of the corresponding 
diffraction-grating, strongly or feebly impressed, according to 
the density of the different parts of the positives. These three 
plates, when superimposed and placed in front of a lens and 
illuminated by a narrow source of light, appear as a correctly 
coloured picture, when viewed with the eye placed in the 
proper position. Perfect registration of the different parts 
of the picture could not be obtained in this way, for obvious 
reasons. I worked for awhile with the thin glass from which 
covers for microscopical slides are made. This gave much 
better results, but was too fragile for practical purposes. It 
then occurred to me that if 1 could get the entire grating- 
system on a single film, not only would the difficulty about 
perfect registration vanish, but the pictures could be repro- 
duced by simple contact-printing on chrom-gelatine plates as 
easily as blue prints are made. I was surprised to find that 
successive exposures of the same plate under the positives, 
perfect registration being secured by marks on the plates, 
produced the desired result. On washing this plate in warm 
water and drying, it becomes the finished coloured photograph. 
Where the reds occur in the original, the spacing of the first 
grating is present; where the yellows occur the spacings of 
both the first and second are to be found superimposed ; where 
the blues occur are the lines of the third grating ; while in 
the white parts of the picture all three spacings are present. 
It seems almost incredible that, by exposing the plate in suc- 
cession under two gratings the spacings, of both should be 
impressed — superimposed — in such a manner as to give the 
colours of each in equal intensity ; but such is the feet. Thus 
far I have had at my disposal but two gratings of only ap- 
proximately the right spacing, one giving the red, the other 
the green : with these I have photographed stained- glass 
windows, birds, and butterflies, and other still-life objects, 
the finished pictures showing reds, yellows, and greens in a 
most beautiful manner. By making a separate plate from 
the blue positive, using the same spacing as with the green, 
and setting this plate behind the other at an angle, 1 have 
obtained the blues and whites, the grating-space being dimin- 
ished by foreshortening, though, of course, perfect regis- 

370 Mr. R. W. Wood on an Application of 

tration of the different portions of the picture could not be 

One of the great advantages of this method is the facility 
with which duplicates can be made. If we place the finished 
picture in a printing-frame over a glass plate coated with 
bichromated gelatine, and expose it to sunlight, on washing 
the plate in warm water we obtain, by a single printing 
process, a second colour-photograph, equal to the first in every 
respect, and also positive. From this second copy we can 
print others, all being positive. 

The apparatus for viewing the pictures consists of a cheap 
double-convex lens mounted on a little frame, as shown in 
fig. 1, with a perforated screen for bringing the eye into the 

right position. I find that, by using a lens of proper focus, 
it is possible to so adjust the apparatus that the picture can be 
seen in its natural colours with both eyes simultaneously, 
since corresponding overlapping spectra are formed on each 
side of the central direct image. A gas- flame turned edge- 
wise, or the filament of an incandescent light, makes a con- 
venient source of light. The colours are of great brilliancy 
and purity, almost too brilliant in fact, though dark reds and 
ochres are reproduced with considerable fidelity. The pictures 
can be projected by employing a powerful arc-light, placing 
a rather wide slit in the overlapping spectra, and mounting 
the projecting lens beyond this. The pictures that I have 
obtained thus far measure 2'5 in. by 2*5 in., and have been 
thrown up about 3 feet square. The fact that only a small 
percentage of the light is utilized makes great amplification 
difficult. Certain experiments that I have made lead me to 
believe that the process can be greatly simplified. 

I have exposed an ordinary photographic plate in a camera 

the Diffraction- Grating to Colour- Photography, 371 

under a diffraction-grating placed in front of, and in contact 
with, the film. On development, we obtain a negative the 
dark portions of which are broken up into fine lines, corre- 
sponding to the lines of the grating; and on viewing this in 
the apparatus just described, the blue components of the picture 
are seen, though not so brilliant as with the transparent 
gelatine plate owing to the coarseness of the grain. 

I believe that by the use of a suitable photographic plate to 
be exposed in succession in the camera, under red, green, and 
blue screens, on the surfaces of which suitable diffraction- 
gratings have been photographed, it will be possible to obtain 
the colour-photograph directly. The screens can be swuno- 
into position in succession by a suitable mechanical arrange- 
ment operated outside of the camera. The plate, on deve- 
lopment, will be a negative in the ordinary sense of the term, 
though when seen in the viewing-apparatus it will appear as 
a coloured positive, since on the transparent portions which 
correspond to black in the original, no grating- lines have 
been impressed : consequently these portions will appear dark. 
The dark portions, however, where the lines are impressed will 
light up in their appropriate colours. From this plate as 
many copies as are desired can be made by contact-printino- 
on bichromated gelatine. 

Of course it is a question whether superimposed gratino-s 
can be impressed on a plate in this manner. Judging from 
the experiments I have made, I imagine that the gratings on 
the colour-screens would have to be made with the opaque 
portions broad in proportion to the transparent. 

I have overcome the difficulty of obtaining large diffraction- 
gratings by building up photographic copies in the followino- 
manner. The original grating ruled on glass was mounted 
against a rectangular aperture in a vertical screen, the lines 
of the grating being horizontal. Immediately below this was 
placed a long piece of heavy plate-glass, supported on a slab of 
slate to avoid possible flexure. A strip of glass, a little wider 
than the grating, sensitized with bichromated gelatine was 
placed in contact with the lines of the grating, and held in 
position by a brass spring. The lower edge of the strip rested 
upon the glass plate so that it could be advanced parallel to 
the lines of the grating, and successive impressions taken by 
means of light coming through the rectangular aperture. In 
this way I secured a long narrow grating; and by mounting 
this against a vertical rectangular aperture, and advancing a 
second sensitized plate across it in precisely the same manner, 
1 obtained a square grating of twenty-five times the area of 
the original. It was in this manner that I prepared the 

e72 Dr. G. Johnstone Stoney on 

orating used to print the impressions on the three positives. 
So well did they perform, that it seemed as if it might be 
possible in this way to build up satisfactory gratings of large 
size for spectroscopic work. Starting with a 1-inch grating 
of 2000 lines, I have bailt up a grating 8 inches square, 
which, when placed over the object-glass of a telescope, 
showed the dark bands in the spectrum of Sirius with great 
distinctness. No especial precautions, other than the use of 
the flat glass plate, were taken to insure absolute parallelism 
of the lines, and I have not Lid time to thoroughly test the 
grating. The spectra, however, are of extraordinary brilliancy; 
and on the whole the field seems promising. This matter will, 
however, be deferred to a subsequent paper. 

Physical Laboratory of the 
University of Wisconsin, Madison. 

XXXIII. Denudation and Deposition. 
By G. Johnstone Stoney, M.A., D.Sc, F.R.S* 

IN a lecture to the Royal Geographical Society, of which a 
copious extract is given in ' Nature ' of the 2nd of 
February, 1899, Dr. J. W. Gregory discusses many of the 
causes which may have led to the existing form of the earth. 
But there is one important factor in the problem left unnoticed, 
namely, the conspicuous alterations of level which may be 
attributed to the earth's compressibility, and which seem to 
have been brought about wherever either denudation or 
deposition have continued over wide areas and for a long time. 

Dr. Gregory makes a convenient division of the earth into 
three parts : — (1) the unknown internal centrosphere ; (2) the 
rocky crust or litho sphere ; (3) the oceanic layer, or hydro- 
sphere. These, with the atmosphere, which may be added as 
a fourth part, make up the whole earth. 

If now we imagine a pyramid whose base is a square centi- 
metre of the surface of the solid part of the earth and whose 
vertex is the earth's centre, it has a volume of about 212 
cubic metres, which is the same as 212 millions of cubic centi- 
metres. This pyramid passes first through the lithospheric 
yhell, or outer crust, and then halfway across the centrosphere 
to the centre of the earth. All the materials of which it 
consists are compressible. Those which lie within the outer 
shell consist mainly of carbonates, silicates, and aluminates, 
a id have probably a coefficient of compressibility about equal 
to that of glass ; while the compressibility of the centrosphere 
is unknown, and may be either more or less. The observed 
form of the earth's surface seems to suggest that the average 
* Communicated by the Author. 

Denudation and Deposition. 373 

compressibility of the lithosphere and centrosphere taken 
together is not far from that of the more incompressible kinds 
of glass. Glass of this description yields to compression 
about 2^ times more than solid cast iron, but less than 
mercury (which seems to be the only liquid metal that has 
been experimented on) in the ratio of 2 to 3. It is about 
20 times more incompressible than water. 

We shall then, as a provisional hypothesis, assume that the 
earth has the same compressibility as the more resistant kinds 
of glass, which lose about 2J billionths of their volume for 
each pressure of a dyne per square centimetre over their 
surface. Combining this with the volume of the earth- 
pyramid given above, we find that our hypothesis leads to the 
conclusion that if the sides of the pyramid were kept from 
yielding, and if the weight of a cubic centimetre of water 
were placed on its outer end, this would reduce its hulk by 
half a cubic centimetre. A cubic centimetre of stone, of 
specific gravity 3, would accordingly depress its outer end by 
1-J- centimetres. It follows from this that if meteors rained 
upon the earth (supposed to be without an ocean) producing 
a deposit over its whole surface a centimetre thick, and of 
material as dense as stone., the result would be that the earth 
after this accession would be smaller ; its surface would sink 
down about half a centimetre. Correspondingly, if by any 
agency a centimetre of the earth's crust could be removed over 
the whole earth, the earth's surface would stand J a centimetre 
higher than before. These are the effects which deposition and 
denudation would respectively produce if they could operate 
over the whole earth. And, if they operate over any extensive 
area of the earth's surface ; they will produce effects of the same 
kind, complicated a little by the displacement of the earth's 
centre of attraction, or rather locus of centres of attraction. 

This may be well seen in the oldest parts of the oldest 
continents — parts of Asia and Africa — to whose present 
elevation denudation *, operating over an extensive area and 
for long ages, has probably chiefly contributed. And, corre- 
spondingly, there is a deepening of those parts of the ocean 
where the deposition of sufficiently heavyt material has been 
going on over a great area for an immense time. 

* Underground waters produce the same dynamical eifect as surface 
denudation, by reason of the materials they remove in solution. 

f Yv^here the sub-aqueous deposit is spread over only a small part of 
the surface of the globe (which is the only case we need consider), the 
compression is due, not to the whole weight of the deposit, but only to 
its excess over the weight of an equal bulk of water. Hence to produce 
an equivalent effect the material must be denser than it would need to 
be if the deposition had been on land. 

374 On Denudation and Deposition. 

The extent of the area is an essential condition, L e., the 
lateral dimensions of the inverted pyramid which has the area 
for its base and the centre of the earth for its vertex. If the 
area is small or narrow, oblique forces exerted by the parts 
surrounding this pyramid come more into play. They enable 
the part within the pyramid to act like a bridge ; and the 
support thus given enables denudation, if limited to a small 
area, to scoop out valleys, and deposition to produce ridges, 
as may be seen in the glaciers and moraines of mountainous 
countries. On the other hand, if the erosion due to glacial 
action takes effect over a great stretch of country, as it does 
in Greenland, and as it formerly did in Ireland, it causes the 
surface to rise. 

A nearly even balance between the two opposite tendencies 
may be seen in Egypt, where borings exhibit fluviatile de- 
posits at great depths below the present surface, although the 
surface is only about as much raised above the sea now as it 
was when those ancient deposits were laid down by the Nile. 
Each year's deposit makes the surface go down, but only 
about as much as its own thickness, so that the new surface 
each year is not far from being at the same level as that of 
the preceding year. If the deposit had taken place over a 
much greater breadth of country, the whole would have gone 
down. It would have become a ridge if it had been confined 
to a much narrower strip and if the river could have been 
kept from diverging. 

A similarly instructive case is that of Brazil, where an 
immense plateau is continuously being denuded by the vast 
rivers that drain it. But here there is also an equally un- 
interrupted addition to the solid materials of the earth by the 
luxuriant tropical vegetation which everywhere prevails ; and 
it is probably because the accessions and withdrawals are 
nearly equal to one another, that the level of the surface has 
been but little changed. 

Denudation may cause the surface to rise within a space 
which is in a considerable degree more circumscribed than 
the areas of elevation hitherto considered, if the conditions 
are such that the stresses that come into existence round the 
boundary of this limited space can produce faults, and pre- 
vent the material which is outside the pyramid from being in 
a position to help to keep down the material which is within. 
This seems to have happened in the case of that vast mass of 
mountains — the Himalayas, the Hindu Kush, and their asso- 
ciated ranges — where excessive denudation accompanied by 
the isolation secured by faults has occasioned a proportionately 
great elevation above what was probably a humble beginning ; 

On Transmission of Light through an Atmosphere. old 

where the deposits in the Bay of Bengal are probably the 
cause of its great depth; and where earthquakes in the 
intervening regions betray when the faults are establishing 
themselves which render the rising and the descending areas 
independent of one another, and allow the denudation on the 
one side and the deposition on the other to produce each its 
full effect, without mutual interference. 

Of course all compressions and dilatations must be accom- 
panied by other movements Avithin the earth, and at all 
depths ; which may be slow but are no less sure. In fact, 
there is no material which can resist yielding to differences 
of pressure, however feeble, if they act for a long time and 
over a large surface ; and such pressures, urging in various 
directions, must arise both from the compressions and dilata- 
tions spoken of above, and from other causes, among which 
movements of heat and the heterogeneous character of the 
materials of which the earth consists are prominent. The 
earth, therefore, is in a state of never-ending change, which 
to become conspicuous to man would only need to be placed 
in some kind of kinematograph arrangement which would 
hurry over millions of years in fractions of a second. These 
effects mix with and complicate those which have been taken 
account of in the present paper. 

It is interesting to note how the agencies we have been 
considering would operate upon other bodies of the universe. 
Events equivalent to denudation and deposition which cause 
excessively slow movements in our small earth, would act w T ith 
increased promptness upon such great planets as Jupiter, 
Saturn, Uranus, and Neptune, and with violence upon bodies 
that attain the size of the sun and stars. On the other 
hand, on bodies with the dimensions of the moon they are 
relatively feeble, and must be very slow in producing any 
appreciable effect. 

XXXIV. On the Transmission of Light through an Atmosphere 
containing Small Particles in Suspension, and on the Origin 
of the Blue of the Sky. By Lord Rayleigh, F.R.S* 

rpHIS subject has been treated in papers published many 
J- years ago |. I resume it in order to examine more 
closely than hitherto the attenuation undergone by the 
primary light on its passage through a medium containing 
small particles, as dependent upon the number and size of the 
particles. Closely connected with this is the interesting 

* Communicated by the Author. 

t Phil. Mag. xli. pp. 107, 274, 447 (1871) ; xii. p. 81 (1881). 

376 Lord Rayleigh on the Transmission of Light through 

question whether the light from the sky can be explained by 
diffraction from the molecules of air themselves, or whether it 
is necessary to appeal to suspended particles composed of 
foreign matter, solid or liquid. It will appear, I think, that 
even in the absence of foreign particles we should still have a 
blue sky *. 

The calculations of the present paper are not needed in 
order to explain the general character of the effects produced. 
In the earliest of those above referred to I illustrated by 
curves the gradual reddening of the transmitted light by 

* My attention was specially directed to tins question a long while ao'o 
by Maxwell in a letter which I may be pardoned for reproducing here. 
Under date Aug. 28, 1873, he wrote :— 

" I have left your papers on the light of the sky, &c. at Cambridge, 
and it would take me, even if I had them, some time to get them assimi- 
lated sufficiently to answer the following question, which I think will 
involve less expense to the energy of the race if you stick the data into 
your formula and send me the result 

" Suppose that there are N spheres of density p and diameter s in unit 
of volume of the medium. Find the index of refraction of the compound 
medium and the coefficient of extinction of light passing through it. 

"The object of the enquiry is, of course, to obtain data about the size 
of the molecules of air. Perhaps it may lead also to data involving the 
density of the aether. The following quantities are known, being com- 
binations of the three unknowns, 

M = mass of molecule of hydrogen ; 

N= number of molecules of any gas in a cubic centimetre at G° C. 

and 760 B. 
s = diameter of molecule in any gas: — 

Known Combinations. 
MN = density. 
Ms 2 from diffusion or viscosity. 

Conjectural Combination. 

— - = density of molecule. 

" If you can give us (i.) the quantity of light scattered in a given 
direction by a stratum of a certain density and thickness ; (ii.) the 
quantity cut out of the direct ray ; and (iii.) the effect of the molecules 
on the index of refraction, which I think ought to come out easily, we 
might get a little more information about these little bodies. 

" You will see by ' Nature,' Aug. 14, 1873, that I make the diameter 
of molecules about ^Vo °f a wave-length. 

" The enquiry into scattering must begin by accounting for the great 
observed transparency of air. I suppose we have no numerical data 
about its absorption. 

"But the index of refraction can be numerically determined, though 
the observation is of a delicate kind, and a comparison of the result 
with the dynamical theory may lead to some new information." 

Subsequently he wrote, " Your letter of Nov. 17 quite accounts for the 
observed transparency of any gas." So far as I remember, my argument 
was of a general character onfy. 

an Atmosphere containing Small Particles in Suspension. 377 

which we see the sun a little before sunset. The same 
reasoning proved, of course, that the spectrum of even a 
vertical sun is modified by the atmosphere in the direction of 
favouring the waves of greater length. 

For such a purpose as the present it makes little difference 
whether we speak in terms of the electromagnetic theory or 
of the elastic solid theory of light ; but to facilitate compari- 
son with former papers on the light from the sky, it will be 
convenient to follow the latter course. The small particle of 
volume T is supposed to be small in all its dimensions in 
comparison with the wave-length (X), and to be of optical 
density D' differing from that (D) of the surrounding 
medium. Then, if the incident vibration be taken as unity, 
the expression for the vibration scattered from the particle in 
a direction making an angle 9 with that of primary vibra- 
tion is 

— ^— _ S in cos — (bt-r)*, . . . (1) 

r being the distance from T of any point along the secondary ray. 
In order to find the whole emission of energy from T we 
have to integrate the square of (1) over the surface of a 
sphere of radius r. The element of area being 2wr 2 sin 6d0, 
we have 

^ S ^2nr*sm6d0 = ±7rP\m*ddd= ~; 

so that the energy emitted from T is represented by 

8tt s (D'-D) 2 T* f9 . 

"3" L>* \^ {Z) 

on such a scale that the energy of the primary wave is unity 
per unit of wave-front area. 

The above relates to a single particle. If there be n 
similar particles per unit volume, the energy emitted from a 
stratum of thickness dee and of unit area is found from (2) by 
introduction of the factor ndx. Since there is no waste of 
energy on the whole, this represents the loss of energy in the 
primary wave. Accordingly, if E be the energy of the pri- 
mary wave, 

lc/E_ 87r 3 n (D'-D) g T a 

e dx ~ a w \ 4 ; • • • ( °) 

whence E = E 6- fe , (I) 

where ; _ 8tt^ (D'-D) 2 T^ 

8 D 9 - ~\ 4 ' ' ' * ^ 

* The factor n was inadvertently omitted in the original memoir. 

378 Lord Rayleigh on the Transmission of Light through 

If we had a sufficiently complete expression for the scattered 
light, we might investigate (5) somewhat more directly by 
considering the resultant of the primary vibration and of 
the secondary vibrations which travel in the same direction. 
If, however, we apply this process to (1), we find that it fails 
to lead us to (5), though it furnishes another result of interest. 
The combination of the secondary waves which travel in the 
direction in question have this peculiarity, that the phases 
are no more distributed at random. The intensity of the 
secondary light is no longer to be arrived at by addition of 
individual intensities, but must be calculated with considera- 
tion of the particular phases involved. If we consider a 
number of particles which all lie upon a primary ray, we see 
that the phases of the secondary vibrations which issue along 
this line are all the same. 

The actual calculation follows a similar course to that by 
which Huygens' conception of the resolution of a wave into 
components corresponding to the various 
parts of the wave-front is usually veri- 
fied. Consider the particles which oc- 
cupy a thin stratum dx perpendicular 
to the primary ray x. Let AP (fig. 1) 
be this stratum and the point where 
the vibration is to be estimated. If 
AF = p, the element of volume is 
dx . "27rpdp, and the number of particles 
to be found in it is deduced by intro- 
duction of the factor n. Moreover, if 

Fisr 1. 

OP = 

AO = 


= x 2 -\-p 2 , and 

pdp = rdr. The resultant at of all the 
secondary vibrations which issue from 
the stratum dx is by (1), with sin equal to unity 


p» D'-D ttT 


cos — - (bt — r) 2irrdr } 


7 D'-DttT . 2tt,, , 

n ax . — y\ — sm — [bt — x) 

U A A, 


To this is to be added the expression for the primary wave 


itself, supposed to advance undisturbed, viz., cos — (bt — x), 

and the resultant will then represent the whole actual dis- 
turbance at as modified by the particles in the stratum dx. 
It appears, therefore, that to the order of approximation 
afforded by (1) the effect of the particles in dx is to modify 
the phase, but not the intensity, of the light which passes 

an Atmosphere containing Small Particles in Suspension. 379 
them. If this be represented by 

cos ^ (fa- a- 8), (7) 


8 is the retardation due to the particles, and we have 

8 = wT^(D / -D)/2D (8) 

If fju be the refractive index of the medium as modified by 
the particles, that of the original medium being taken as 
unity, b s =({i — l)dw, and 

At -l = nT(D , -D)/2D (9) 

If /j! denote the refractive index of the material composing 
the particles regarded as continuous, D' /D = fi' 2 , and 

/.-l=i«T0i»-l), (10) 

reducing to 

ft-I = «T(/*'-l) (11) 

in the case where yJ — 1 can be regarded as small. 

It is only in the latter case that the formulas of the elastic- 
solid theory are applicable to light. In the electric theory, 
to be preferred on every ground except that of easy intelli- 
gibility, the results are more complicated in that when (//— 1) 
is not small, the scattered ray depends upon the shape and 
not merely upon the volume of the small obstacle. In 
the case of spheres we are to replace (D / — D)/D by 
3(K / -K)/(K' + 2K), where K, K' are the dielectric constants 
proper to the medium and to the obstacle respectively*; so 
that instead of (10) 

1 3tiT fS*-l 

^ 1= TAl ^ 

On the same suppositions (5) is replaced by 
On either theory 

( u '2_\ \2 T2 

3nX 4 ' { J 

a formula giving the coefficient of transmission in terms of 
the refraction, and of the number of particles per unit volume. 
We have seen that when we attempt to find directly from 
(1) the effect of the particles upon the transmitted primary 
wave, we succeed only so far as regards the retardation. In 

* Phil. Mag-, xii. p. 98 (1881). For the corresponding theory in the 
case of an ellipsoidal obstacle, see Phil. Mag. vol. xliv. p. 18 (1897). 

380 Lord Rayleigh on the Transmission of Light through 

order to determine the attenuation by this process it would be 
necessary to supplement (1) by a term involving 

but this is of higher order of smallness. We could, however, 
reverse the process and determine the small term in question 
a posteriori by means of the value of the attenuation obtained 
indirectly from (1), at least as far as concerns the secondary 
light emitted in the direction of the primary ray. 

The theory of these effects may be illustrated by a com- 
pletely worked out case, such as that of a small rigid and 
fixed spherical obstacle (radius c) upon which plane waves of 
sound impinge *. It would take too much space to give full 
details here, but a few indications may be of use to a reader 
desirous of pursuing the matter further. 

The expressions for the terms of orders and 1 in spherical 
harmonics of the velocity-potential of the secondary disturbance 
are given in equations (16), (17), § 334. With introduction 
of approximate values of 7 and 7 b viz. 

we get 

[f »] + DM = - % 3 (! + t) cos *(*-') 

+ ^( 1 ~t) sin *( a *-'-)> • • ( 15 ) 

in which c is the radius of the sphere, and k = 2ir/\. This 
corresponds to the primary wave 

1$] = cos k(at + w), (16) 

and includes the most important terms from all sources in the 
multipliers of cos k(at—r), sin k(at -r). Along the course of 
the primary ray (fi= — 1) it reduces to 

k 2 c z lk b c 6 

[^o] + [fi]=-^ cos ^-^)+ -gg^ sin *(o*-r)- • ( 17 ) 

We have now to calculate by the method of FresnePs zones 
the effect of a distribution of n spheres per unit volume. 
We find, corresponding to (6). for the effect of a layer of 
thickness dx, 

1irndx{\ke % sin k{at + «)- 3 7 ^c 6 cos h{at + x)}. . (18) 

To ihis is to be added the expression (16) for the primary 
wave. The coefficient of cos k\at-\-x) is thus altered by the 
* ' Theory of Sound,' 2nd ed. § 334. 

an Atmosphere containing Small Particles in Suspension. 381 

particles in the layer dx from unity to (l—^k 4 c 6 7rndx), and 
the coefficient of s'm k (at + x) from to \kc ?Jr nndx. Thus, if 
E be the energy of the primary wave, 

dE/E=-%k 4 c 6 7rndx; 

so that if, as in (4), E =E e~ hx , 

h = l7rnk 4 c 6 (19) 

The same result may be obtained indirectly from the first 
term of (15). For the whole energy emitted from one sphere 
may be reckoned as 

|£j_W(l+|/.)ty=-^p > . ■ • (20) 

unity representing the energy of the primary wave per unit 
area of wave-front. From (20) we deduce the same value of 
h as in (19). 

The first term of (18) gives the refractivity of the medium. 
If 8 be the retardation due to the spheres of the stratum dx, 

sin k8=^kc d irndXj 

or h = ±Trnc z dx (21) 

Thus, if fi be the refractive index as modified by the spheres, 
that of the original medium being unity, 

^-l=j7rnc 3 = ip, (22) 

where p denotes the (small) ratio of the volume occupied by 
the spheres to the whole volume, This result agrees with 
equations formerly obtained for the refractivity of a medium 
containing spherical obstacles disposed in cubic order*. 

Let us now inquire what degree of transparency of air is 
admitted by its molecular constitution, i. e., in the absence 
of all foreign matter. We may take A=6xl0~ 5 centim., 
fi — 1 = *0003; whence from (14) we obtain as the distance x, 
equal to 1/h, which light must travel in order to undergo 
attenuation in the ratio e : 1 , 

<z>=4-!xl0- 13 xn (23) 

The completion of the calculation requires the value of n. 
Unfortunately this number — according to Avogadro's law 
the same for all gases — can hardly be regarded as known. 
Maxwell f estimates the number of molecules under standard 

* Phil. Mag. vol. xxxiv. p. 499 (1892). Suppose m = oo , o- = oo . 
t " Molecules," Nature, viii. p. 440 (1873). 

Phil. Mag. S. 5. Vol. 47. No. 287. April 1899. 2 D 

382 Lord Rayleigh on the Transmission of Light through 

conditions as 19 X 10 18 per cub. centim. If we nse this value 
of », we find 

# = 8*3xl0 6 cm. = 83 kilometres, 

as the distance through which light must pass through air 
at atmospheric pressure before its intensity is reduced in the 
ratio of 2*7 : 1. 

Although Mount Everest appears fairly bright at 100 miles 
distance as seen from the neighbourhood of Darjeeling, we can- 
not suppose that the atmosphere is ns transparent as is 
implied in the above numbers ; and of course this is not to 
be expected, since there is certainly suspended matter to be 
reckoned with. Perhaps the best data for a comparison are 
those afforded by the varying brightness of stars at various 
altitudes. Bouguer and others estimate about *8 for the 
transmission of light through the entire atmosphere from a 
star in the zenith. This corresponds to 8' 3 kilometres of air 
at standard pressure. At this rate the transmission through 
83 kilometres would be ("8) 10 , or "11, instead of 1/e or *37. 
It appears then that the nctual transmission through 83 kilo- 
metres is only about 3 times less than that calculated (with 
the above value of n) from molecular diffraction without any 
allowance for foreign matter at all. And we may conclude 
that the light scattered from the molecules would suffice to 
give us a blue sky, not so very greatly darker than that 
actually enjoyed. 

If n be regarded as altogether unknown, we may reverse 
our argument, and we then arrive at the conclusion that n 
cannot be greatly less than was estimated by Maxwell. A 
lower limit for n, say 7 X 10 18 per cubic centimetre, is some- 
what sharply indicated. For a still smaller value, or rather 
the increased individual efficacy which according to the 
observed refraction would be its accompaniment, must lead to 
a less degree of transparency than is actually found. When 
we take into account the known presence of foreign matter, 
we shall probably see no ground for any reduction of 
Maxwell's number. 

The results which we have obtained are based upon (14), 
and are as true as the theories from which that equation was 
derived. In the electromagnetic theory we have treated the 
molecules as spherical continuous bodies differing from the 
rest of the medium merely in the value of their dielectric 
constant. If we abandon the restriction as to sphericity, the 
results will be modified in a manner that cannot be precisely 
defined until the shape is specified. On the whole, however, it 
does not appear probable that this consideration would greatly 
affect the calculation as to transparency, since the particles 

an Atmosphere containing Small Partides in Suspension. 383 

must be supposed to be oriented in all directions indifferently. 
But the theoretical conclusion that the light diffracted in a 
direction perpendicular to the primary rays should be com- 
pletely polarized may well be seriously disturbed. If the 
view, suggested in the present paper, that a large part of the 
light from the sky is diffracted from the molecules themselves, 
be correct, the observed incomplete polarization at 90° from 
the Sun may be partly due to the molecules behaving rather 
as elongated bodies with indifferent orientation than as spheres 
of homogeneous material. 

Again, the suppositions upon which we have proceeded 
give no account of dispersion. That the refraction of gases 
increases as the wave-length diminishes is an observed fact ; 
and it is probable that the relation between refraction and 
transparency expressed in (14) holds good for each wave- 
length. If so, the falling off of transparency at the blue end 
of the spectrum will be even more marked than according to 
the inverse fourth power of the wave-length. 

An interesting question arises as to whether (14) can be 
applied to highly compressed gases and to liquids or solids. 
Since approximately (/jl — 1) is proportional to n, so also is 
h according to (14). We have no reason to suppose that 
the purest water is any more transparent than (14) would 
indicate ; but it is more than doubtful whether the calcula- 
tions are applicable to such a case, where the fundamental 
supposition, that the phases are entirely at random, is violated. 
When the volume occupied by the molecules is no longer 
very small compared with the whole volume, the fact that 
two molecules cannot occupy the same space detracts from 
the random character of the distribution. And when, as in 
liquids and solids, there is some approach to a regular spacing, 
the scattered light must be much less than upon a theory of 
random distribution . 

Hitherto we have considered the case of obstacles small 
compared to the wave-length. In conclusion it may not be 
inappropriate to make a few remarks upon the opposite 
extreme case and to consider briefly the obstruction presented, 
for example, by a shower of rain, where tbe diameters of the 
drops are large multiples of the wave-length of light. 

The full solution of the problem presented by spherical 
drops of water w r ould include the theory of the rainbow, 
and if practicable at all would be a very complicated matter. 
But so far as the direct light is concerned, it would seem to 
make little difference whether we have to do with a spherical 
refracting drop, or with an opaque disk of the same diameter. 


384 On Transmission of Light through an Atmosphere. 

Let us suppose then that a large number of small disks are 
distributed at random over a plane parallel to a wave-front, 
and let us consider their effect upon the direct light at a 
great distance behind. The plane of the disks may be divided 
into a system of Fresnel's zones, each of which will by 
hypothesis include a large number of disks. If a be the area 
of each disk, and v the number distributed per unit of area 
of the plane, the efficiency of each zone is diminished in the 
ratio 1: 1 — va, and, so far as the direct wave is concerned, 
this is the only effect. The amplitude of the direct wave is 
accordingly reduced in the ratio 1 : 1— va, or, if we denote the 
relative opaque area by m, in the ratio 1 : 1 — m*. A second 
operation of the same kind will reduce the amplitude to 
(1 — m) 2 , and so on. After x passages the amplitude is 
(1 — m) x , which if m be very small may be equated to e~ mx . 
Here mx denotes the whole opaque area passed, reckoned 
per unit area of wave-front ; and it would seem that the result 
is applicable to any sufficiently sparse random distribution of 

It may be of interest to give a numerical example. If the 
unit of length be the centimetre and x the distance travelled, 
m will denote the projected area of the drops situated in one 
cubic centimetre. Suppose now that a is the radius of a 
drop, and n the number of drops per cubic centimetre, then 
m = mra 2 . The distance required to reduce the amplitude in 
the ratio e : 1 is given by 

x = 1/mra 2 . 

Suppose that a=^ centim., then the above-named reduction 
will occur in a distance of one kilometre (x = 90 b ) when n is 
about 10 -3 , i. e. when there is about one drop of one milli- 
metre diameter per litre. 

It should be noticed that according to this theory a distant 
point of light seen through a shower of rain ultimately become*! 
invisible, not by failure of definition, but by loss of intensity 
either absolutely or relatively to the scattered light. 

* The intemity of the direct wave is 1 — 2?n, and that of the scattered 
light m, making altogether 1—m. 

[ 385 ] 

XXXV. On Opacity, By Professor Oliver Lodge, D.Sc, 
LL.D., PM.S., President of the Physical Society*. 

MY attention has recently been called to the subject of the 
transmission of electromagnetic waves by conducting 
dielectrics — in other words, to the opacity of imperfectly 
conducting material to light. The question arose when an 
attempt was being made to signal inductively through a 
stratum of earth or sea, how far the intervening layers of 
moderately conducting material were able to act as a screen ; 
the question also arises in the transmission of Hertz waves 
through partial conductors, and again in the transparency of 
gold-leaf and other homogeneous substances to light. 

The earliest treatment of such subjects is due of course to 
Clerk Maxwell thirty-four years ago, when, with unexampled 
genius, he laid down the fundamental laws for the propagation 
of electric waves in simple dielectrics, in crystalline media, and 
in conducting media. He also realised there was some strong 
aualogy between the transmission of such waves through space 
and the transmission of pulses of current along a telegraph- 
wire. But naturally at that early date not every detail of the 
investigation was equally satisfactory and complete. 

Since that time, and using Maxwell as a basis, several 
mathematicians have developed the theory further, and no 
one with more comprehensive thoroughness than Mr. Oliver 
Heaviside, who, as I have said before, has gone into these 
matters with extraordinarily clear and far vision. I may 
take the opportunity of calling or recalling to the notice of 
the Society, as well as of myself, some of the simpler develop- 
ments of Mr. Heaviside's theory and manner of unifying 
phenomena and processes at first sight apparently different ; 
but first I will deal with the better-known aspects of the 

Maxwell deals with the relation between conductivity and 
opacity in his Art. 7b>8 and on practically to the end of that 
famous chapter xx , (' Electromagnetic Theory of Light ' ) . He 
discriminates, though not very explicitly or obtrusively, be- 
tween the two extreme cases, (1) when inductive capacity or 
electric inductivity is the dominant feature of the medium — 
when, for instance, it is a slightly conducting dielectric, and 
(2) the other extreme case, when conductivity is the pre- 
dominant feature. 

* Communicated by the Physical Society of London, being the Presi- 
dential Address for 1899. 

386 Dr. Oliver Lodge on Opacity. 

The equation for the second case, that of predominant con- 
ductivity, is 

da? " <r dt> U 

F being practically any vector representing the amplitude of 
the disturbance ; for since we need not trouble ourselves with 
geometrical considerations such as the oblique incidence of 
waves on a boundary &c, we are at liberty to write the y 
merely as d/dx, taking the beam parallel and the incidence 

No examples are given by Maxwell of the solution of this 
equation, because it is obviously analogous to the ordinary 
heat diffusion fully treated by Fourier. 

Suffice it for us to say that, taking F at the origin as 
represented by a simple harmonic disturbance Y =e ipt . the 
solution of equation (1) 

f? = ^?F (!') 

dx 2 a 

is F = F e~^ = 0-Q*+fc«, 

where Q = ^/(M) = ^/&. (1 + i , ; 


F = r(^)Sos(^-(^J,), ... (2) 

an equation which exhibits no true elastic wave propa- 
gation at a definite velocity, but a trailing and distorted 
progress, with every harmonic constituent going at a diffe- 
rent pace, and dying out at a different rate ; in other words, 
the diffusion so well known in the case of the variable stage of 
heat-conduction through a slab. 

In such conduction the gain of heat by any element whose 
heat capacity is cpdx is proportional to the difference of the 
temperature gradient at its fore and aft surfaces, so that 

, dB , _ dO 
r dt dx 

or, what is the same thing, 

da? Tdt> 

the same as the equation (1) above ; wherefore the constant 
cp/k, the reciprocal of the thermometric conductivity, takes the 

Dr. Oliver Lodge on Opacity. 387 

place of 4:7t/jl/o; that is, of electric conductivity; otherwise the 
heat solution is the same as (2). The 4-7T has come in from 
an unfortunate convention, but it is remarkable that the con- 
ductivity term is inverted. The reason of the inversion of 
this constant is that, whereas the substance conveys the heat 
waves, and by its conductivity aids their advance, the aether 
conveys the electric waves, and the substance only screens 
and opposes, reflects, or dissipates them. 

This is the case applied to sea-water and low frequency by 
Mr. Whitehead in a paper which he gave to this Society in 
June 1897, being prompted thereto by the difficulty which 
Mr. Evershed and the Post Office had found in some trials of 
induction signalling at the Goodwin Sands between a coil 
round a ship at the surface and another coil submerged at a 
depth of 10 or 12 fathoms. It was suspected that the con- 
ductivity of the water mopped up a considerable proportion 
of the induced currents, and Mr. Whitehead's calculation 
tended, or was held to tend, to support that conclusion. 

To the discussion Mr. Heaviside communicated what was 
apparently, as reported, a brief statement ; but I learn that in 
reality it w T as a carefully written note of three pages, which 
recently he has been good enough to lend me a copy of. In 
that note he calls attention to a theory of the whole subject 
which in 1887 he had w T orked out and printed in his collected 
' Electrical Papers,' but which has very likely been over- 
looked. It seems to me a pity that a note by Mr. Heaviside 
should have been so abridged in the reported discussion as 
to be practically useless ; and I am permitted to quote it here 
as an appendix (p. 113). 

Meanwhile, taking the diffusion case as applicable to sea- 
water with moderately low acoustic frequency;, we see that the 
induction effect decreases geometrically with the thickness of 
the oceanic layer, and that the logarithmic decrement of the 

amplitude of the oscillation is \/( )> where a is the 

specific resistance of sea-water and pftir is the frequency. 

Mr. Evershed has measured a- and found it 2 x 10 10 C.G.S., 
that is to say 2 x 10 10 fx square centim. per second ; so putting 
in this value and taking a frequency of 16 per second, the 
amplitude is reduced to 1/eth of what its value would have 
been at the same distance in a perfect insulator, by a depth 

/ a /( 2 x 10 1( y \ _ /1(P_10 5 

V 2*/*p V V2irii, x 2tt x 16 / " V 320 18 centllu - 

= 55 metres. 
Four or five times this thickness of intervening sea would 

388 Dr. Oliver Lodge on Opacity. 

reduce the result at the 16 frequency to insignificance (each 
55-metre-layer reducing the energy to \ of what entered it) ; 
but if the frequency were, say, 400 per second instead of 16 
it would be five times more damped, and the damping thick- 
ness (the depth reducing the amplitude in the ratio e : 1) 
would in that case be only eleven metres. 

It is clear that in a sea 10 fathoms (or say 20 metres) 
deep the failure to inductively operate a "call " responding to 
a frequency of 1 6 per second was not due to the screening effect 
of sea-water *. 

Maxwell, however, is more interested in the propagation 
of actual light, that is to say, in waves whose frequency is 
about 5 x 10 u per second ; and for that he evidently does 
not consider that the simple diffusion theory is suitable. It 
certainly is not applicable to light passing through so feeble 
a conductor as salt water. He attends mainly therefore to 
the other and more interesting case, where electric inductive- 
capacity predominates over the damping effect of conduc- 
tivity, and where true waves therefore advance with an 
approximately definite velocity 

though it is to be noted that the slight sorting out of waves 
of different frequency, called dispersion, is an approximation 
to the case of pure diffusion where the speed is as the square 
root of the frequency, and is accompanied, moreover, as it 
ought to be, by a certain amount of differential or selective 

To treat the case of waves in a conductor, the same damping 
term as before has to be added to the ordinary wave equation, 
and so we have 

£-*£+?£ m 

Taking ¥ =e ipt again, it may be written 

g=(-^ + ^)F, .... (*) 
the same form as equation (1 ; ) ; so the solution is again 

Y = e -Q*+ipt } 

* I learn that the ship supporting the secondary cable was of metal, 
and that the primary or submerged cable was sheathed in uninsulated 
metal, viz. in iron, which would no doubt be practically short-circuited 
by the sea-water. Opacity of the medium is in that case a superfluous 
explanation of the failure, since a closed secondary existed close to both 
sending and receiving circuit. 

Dr. Oliver Lodge on Opacity. 389 

with Q 2 equal to the coefficient of F in (3') . Maxwell, however, 
does not happen to extract the square root of this quantity, 
but, assuming the answer to be of the form (for a simply 
harmonic disturbance) [modifying his letters, vol. ii. § 798] 

e~™ cos (pt — qx), 

he differentiates and equates coefficients, thus getting 

q 2 — r 2 = fJbKp 2 , 2rq= — , 

as the conditions enabling it to satisfy the differential equa- 
tion. This of course gives for the logarithmic decrement, 
or coefficient of absorption, 

2jrji .p 

? * 
p/q being precisely the velocity of propagation of the train of 

waves. Though not exactly equal- to 1/ V/llK, the true velo- 
city of wave propagation, except as a first approximation, in 
an absorbing medium, yet practically this velocity p/q or X/T 
is independent of the frequency except in strongly absorbent 
substances where there are dispersional complications ; and 
so the damping is_, in simple cases, practically independent 
of the frequency too. 

With this simple velocity in mind Maxwell proceeds to 
apply his theory numerically to gold-leaf, calculating its 
theoretical transparency, and finding, as every one knows, 
that it comes out discordant with experiment, being out of 
all comparison * smaller than what experiment gives. 

But then it is somewhat surprising to find gold treated as 
a substance in which conductivity does not predominate over 
specific inductive capacity. 

The differential equation is quite general and applies to 
any substance, and since the solution given is a true solu- 
tion, it too must apply to any substance when properly 
interpreted ; but writing it in the form just given does not 
suggest the full and complete solution. It seems to apply 
only to slightly damped waves, and indeed, Maxwell seems to 
consider it desirable to rewrite the original equation with omis- 
sion of K, for the purpose of dealing with good conductors. 

By a slip, however, he treats gold for the moment as if it 
belonged to the category of poor conductors, and as if ab- 
sorption in a thickness such as gold-leaf could be treated as 
a moderate damping of otherwise progressive waves. 

* The fraction representing the calculated transmission by a film half 
a wave thick has two thousand digits in its denominator : see below. 

390 Dr. Oliver Lodge on Opacity. 

The slip was naturally due to a consideration of the 
extreme frequency of light vibrations ; but attention to the 
more complete expression for the solution of the same differ- 
ential equation, given in 1887 by Mr. Heaviside and quoted 
in the note to this Society above referred to, puts the matter 
in a proper position. Referring to his ( Electrical Papers,' 
vol. ii. p. 422, he writes down the general value of the 
coefficient of absorption as follows (translating into our 


without regard to whether the conductivity of the medium is 
large or small ; where v is the undamped or true velocity of 
wave propagation in the medium ({aK)~K 

Of course Maxwell could have got this expression in an instant 
by extracting the square root of the quantity Q, the coefficient 
of F in equation (o f ) written above. I do not suppose that 
there is anything of the slightest interest from the mathe- 
matician's point of view, the interest lies in the physical 
application ; but as this is not a mathematical Society it is 
permissible, and I believe proper, to indicate steps for the 
working out of the general solution of equation (3) by extract- 
ing the square root of the complex quantity Q. 

The equation is 

and the solution is 

Q = ^/-^K/ + ^^ =* + ;/? say. 

Squaring we get, just as Maxwell did, 

Squaring again and adding 

(a* + f3 2 ) 2 = (a 2 - /3 2 ) 2 + 4a 2 /3 2 = fi 2 Ky + ]^fjpt 



^-v{. + (^)*}! 

2/ 3.= P K i ,.{ v /(l+(i^)*) + l}, . . (4) 


Dr. Oliver Lodge on Opacity. 391 

and 2a 2 = the same with the last sign negative, 

^(^[{i+^yy-i]*. . . ( 5) 

which is the logarithmic decrement of the oscillation per 
unit of distance, or the reciprocal of the thickness which 
reduces the amplitude in the ratio 1 : e (or the energy to -f) 
of the value it would have at the same place without 

Using these values for a and ft, the radiation-vector in 
general, after passing through any thickness x of any medium 
whose magnetic permeability and other properties are con- 
stant, is 

F = F £- aX cos(^- / &z'), (6) 

the speed of advance of the wave-train being pi ft. 

Now not only the numerical, value but the form of this 
damping constant a, depends on the magnitude of the nume- 


rical quantity — t?, which may be called the critical number*, 
and may also be written 

p*K/K > ••••••• A<) 

where K, the absolute specific inductive capacity of the 
medium, is replaced by its relative value in terms of K for 

vacuum, and by — .=the velocity of light in vacuo =v . 

Now for all ordinary frequencies and good conductors this 
critical number is large ; and in that case it will be found that 

and that ft is identically the same. This represents the 
simple diffusion case, and leads to equation (2). 

On the other hand, for luminous frequency and bad con- 
ductors, the critical quantity is small, and in that case 

* An instructive mode of writing a and /3 in general is given in (11") 
or (12") below, where the above critical number is called tan e : — 

av Vcose = p sin|e, 
fiv Vcos e =■ p cos y. 

392 Dr. Oliver Lodge on Opacity. 


giving the solution 

F=F ,--^cos j p(j-^-). . . (8) 

This expresses the transmission of light through imperfect 
insulators, and is the case specially applied by Maxwell to cal- 
culations of opacity. Its form serves likewise for telegraphic 
signals or Hertz waves transmitted by a highly-conducting 
aerial wire ; the damping, if any, is independent of frequency 
and there is true undistorted wave-propagation at velocity 
v = 1/ VLB ; the constants belonging to unit length of the wire. 
The current (or potential) at any time and place is 

iu- — 

C = C e-2Lv cosp(t— x vLS). ... (9) 

The other extreme case, that of diffusion, represented by 
equation (2), is analogous to the well-known transmission of 
slow signals by Atlantic cables, that is by long cables where 
resistance and capacity are predominant, giving the so-called 
KR law (only that I will write it RS), 

C = C oe - 7(l?,RS)x cos{^-v(^RS) ( r}; . . (10) 

wherefore the damping distance in a cable is 



Thus, in comparing the cable case with the penetration of 
waves into a conductor and with the case of thermal con- 
duction, the following quantities correspond : 

2W 2o- ? 2k' 

cp is the heat-capacity per unit volume, S is the electric 
capacity per unit length ; k is the thermal conductivity per 
unit volume, 1/R is the electric conductance per unit length. 
So these agree exactly ; but in the middle case, that of waves 
entering a conductor, there is a notable inversion, representing 
a real physical fact. 4z7Tfju may be called the density and may 
be compared with p or with 1/S, that is with elasticity-hv 2 ; 
but a is the resistance per unit volume instead of the con- 
ductance. The reason of course is that whereas good con- 
ductivity helps the cable-signals or the heat along, it by no 
means helps the waves into the conductor. Conductivity aids 

Dr. Oliver Lodge on Opacity* 393 

their slipping along the boundary of a conductor, but it 
retards their passing across the boundary and entering a 
conductor. As regards waves entering a conductor, the 
effect of conductivity is a screening effect, not a trans- 
mitting effect, and it is the bad conductor which alone has 
a chance of being a transparent medium. 

It may be convenient to telegraphists, accustomed to 
think in terms of the " KR-law " and comparing equa- 
tions (2) and (10), to note that the quantity 4c7t/jl/<t — that 
is, practically, the specific conductivity in electromagnetic 
measure (multiplied by a meaningless 47T because of an 
unfortunate initial convention) — takes the place of KR (i. e. of 
RS), but that otherwise the damping-out of the waves as 
they enter a good conductor is exactly like the damping-out of 
the signals as they progress through a cable ; or again as elec- 
trification travels along a cotton thread, or as a temperature 
pulse makes its way through a slab ; and yet another case, 
though it is different in many respects, yet has some simi- 
larities, viz. the ultimate distance the melting-point of wax 
travels along a bar in Ingenhousz's conductivity apparatus, — - 
the same law of inverse square of distance for effective reach 
of signal holding in each case. 

Now it is pointed out by Mr. Heaviside in several places 
in his writings that, whereas the transmission of high- 
frequency waves by a nearly transparent substance corre- 
sponds by analogy to the conveyance of Hertz waves along 
aerial wires (or along cables for that matter, if sufficiently con- 
ducting) , and whereas the absorption of low-frequency waves 
by a conducting substance corresponds, also by analogy, to the 
diffusion of pulses along a telegraph-cable whose self-induction 
is neglected — its resistance and capacity being prominent, — 
the intermediate case of waves of moderate frequency in a 
conductor of intermediate opacity corresponds to the more 
general cable case where self-induction becomes important and 
where leakage also must be taken into account ; because it is 
leakage conductance that is the conductance of the dielectric 
concerned in plane waves. This last is therefore a real, and 
not only an analogic, correspondence. 

Writing R : Si L : Q 1 for the resistance, the capacity (" per- 
mittance"), the inductance, and the leakage-conductance 
(" leakance ") respectively, per unit length, the general 
equations to cable-signalling are given in Mr. Heaviside's 
Electromagnetic Theory ' thus : — 

at ax at dec 

394 Dr. Oliver Lodge on Opacity. 

or for a simple harmonic disturbance, 

g= (B,+yL0(Q 1 + vS,)V . . . (11) 
= (*+WY, 

whose solution therefore is 

V = V e~ ax cos (pt-px) *...". (11') 

There are several interesting special cases : — 

The old cable theory of Lord Kelvin is obtained by omitting 
both Q and L; thus getting equation (2). 

The transmission of Hertz waves along a perfectly-con- 
ducting insulated wire is obtained by omitting Q and R ; the 
speed of such transmission being 1/^(L 1 S 1 ). Resistance 
in the wire brings it to the form (9), where the damping- 
depends on the ratio of the capacity constant RS to the self- 
induction constant L/S ; because the index R/2Li; equals half 
the square root of this ratio ; but it must be remembered 
that R has the throttled value due to merely superficial 
penetration. The case is approximated to in telephony 

A remarkable case of undistorted (though attenuated) 
transmission through a cable (discovered by Mr. Heaviside, 
but not yet practically applied) is obtained by taking 

R/L = Q/S = r ; 
the solution being then 

rx / T \ 


due to fit) at #=0. All frequencies are thus treated alike, 
and a true velocity of transmission makes its reappearance. 
This is what he calls his ''distortionless circuit," which may 
yet play an important part in practice. 

And lastly, the two cases which for brevity may be 
treated together, the case of perfect insulation, Q = 0, on 
the one hand, and the case of perfect wire conduction, R=0, 
on the other. For either of these cases the general expres- 

.W =te .L4{ I + (f L )*}'{ 1+ (J)'}'-{J?S-'}] 

* I don't know whether the following simple general expression for 
a and /3 has been recorded by anyone : writing E/j9L=tan e and Q/^S = 
tan e', 

' p sin or cos K*+0 nt'n\ 

a or £ = L - . — 2 ' J , (11 ) 

v (cose cos e)i 

which is shorter than (12). 


Dr. Oliver Lodge on Opacity. 395 

becomes exactly of the form (4) or (5) reckoned above for the 
general screening-effect, or opacity, of conducting media in 

For the number which takes the place of the quantity there 
called the critical number, namely either R//>L or Q/pS, the 
other being zero, we may write tan e ; in which case the 
above is 

«»ori8»=J|>»L 1 S 1 (sece+l); (12') 

or, rewriting in a sufficiently obvious manner, with 2tt/\ for 
p/v if we choose, 

_^sini6 pcos±e ^ ff 

v{qos e)*' v(cose)a 

Instead of attending to special cases, if we attend to the 
general cable equation (11) as it stands, we see that it is 
more general than the corresponding equation (3) to waves 
in space, because it contains the extra possibility R of wire 
resistance, which does not exist in free space. 

Mr. Heaviside, however, prefers to unify the whole by the 
introduction of a hypothetical and as yet undiscovered dissipa- 
tion-possibility in space, or in material bodies occupying 
space, which he calls magnetic conductance, and which, 
though supposed to be non-existent, may perhaps conceivably 
represent the reciprocal of some kind of hysteresis, either the 
electric or the magnetic variety. Calling this g, (#H 2 is to 
be the dissipation term corresponding with RC 2 ) , the equation 
to waves in space becomes 

V 2 F=(« ? + y. / a)(^ + ^K)F, . . . (13) 

just like the general cable case. And a curious kind of 
transparency, attenuation without distortion, would belong to 
a medium in which both conductivities coexisted in such 
proportion that g : ii = Airk : K ; for g would destroy H just 
as k destroys E. 

In the cable, F may be either current or potential, and 
LSv 2 = l. In space, F may be either electric or magnetic 
intensity, and /jlKv 2 =1 ; but observe that g takes the place 
not of Q but of R, while it is Att/ct that takes the place of Q. 
Resistance in the wire and electric conductivity in space 
do not produce similar effects. If there is any analogue 
in space to wire resistance it is magnetic not electric con- 

The important thing is of course that the wire does not 
convey the energy but dissipates it, so that the dissipation by 
wire-resistance and the dissipation by space-hysteresis to 

396 Dr. Oliver Lodge on Opacity. 

that extent correspond. The screening effect of space- 
conductivity involves the very same dielectric property as that 
which causes leakage or imperfect insulation of the cable 

Returning to the imaginary magnetic conductivity, let 
us trace what its effects would be if it existed, and try to 
grasp it. It effect would be to kill out the magnetism of 
permanent magnets in time, and generally to waste away the 
energy of a static magnetic field, just as resistance in w T ires 
wastes the energy of an unmaintained current and so kills 
out the magnetism of its field. I spoke above as if it were 
conceivable that such magnetic conductivity could actually 
in some degree exist, likening it to a kind of hysteresis ; but 
hysteresis — the enclosure of a loop between a to and fro path 
— is a phenomenon essentially associated with fluctuations, and 
cannot exist in a steady field with everything stationary. 
Admitted : but then the molecules are not stationary, and the 
behaviour of molecules in the Zeeman and Righi phenomena, 
or still more strikingly in the gratuitous radiations discovered 
by Edmond Becquerel, and more widely recognized by others, 
especially by Monsieur et Madame Curie, (not really gratuitous 
but effected probably by conversion into high-pitched radiation 
of energy supplied from low-pitched sources), — the way 
molecules of absorbent substances behave, seems to render 
possible, or at least conceivable, something like a minute 
magnetic conductivity in radiative or absorptive substances. 
Mr. Heaviside, however, never introduced it as a physical 
fact for which there was any experimental evidence, but as a 
physical possibility and especially as a mathematical auxiliary 
and unifier of treatment, and that is all that we need here 
consider it to be ; but we may trace in rather more detail its 
effect if it did exist. 

Suppose the magnetism of a magnet decayed, what would 
happen to its lines of force ? They would gradually shrink 
into smaller loops and ultimately into molecular ones. The 
generation of a magnetic field is always the opening out of 
previously existing molecular magnetic loops ; there is no 
such thing as the creation of a magnetic field, except in the 
sense of moving it into a fresh place or expanding it over a 
wider region *. So also the destruction of a magnetic field 
merely means the shrinkage of its lines of force (or lines of 
induction, I am not here discriminating between them). 
Now consider an electric current in a wire : — a cylindrical 
magnetic field surrounds it, and if the current gradually de- 
creases in strength the magnetic energy gradually sinks into 
* This may be disagreed with. 

Dr. Oliver Lodge on Opacity. 397 

the wire as its lines slowly collapse. But observe that the 
electric energy of the field remains unchanged by thi3 
process : if the wire were electrostatically charged it would 
remain charged, its average potential can remain constant. 
Let the wire for instance be perfectly conducting, then the 
current needs no maintenance, the potential might be 
uniform (though in general there would be waves running to 
and fro), and both the electric and magnetic fields continue 
for ever, unless there is some dissipative property in space. 

Two kinds of dissipative property may be imagined in 
matter filling space : first, and most ordinary, an electric con- 
ductivity or simple leakage, the result of which will be to 
equalize the potential throughout space and destroy the electric 
field, without necessarily affecting the magnetic Held, and so 
without stopping the steady circulation of the current mani- 
fested by that field. The other dissipativo property in space 
that could be imagined would be magnetic conductivity ; the 
result of which would be to shrink all the circular lines of 
magnetic force slowly upon the wire, thus destroying the 
magnetic field, and with it (by the circuital relation) the 
current ; but leaving the electrostatic potential and the electric 
field unchanged. And this imaginary effect of the medium 
in surrounding space is exactly the real effect caused by what 
is called electric resistance in the wire *. 

Now for a simply progressive undistorted wave, i. e. one 
with no character of diffusion about it, but all frequencies 
travelling at the same quite definite speed l/\/yu,K,it is essential 
that the electric and magnetic energies shall be equal. If 
both are weakened in the same proportion, the wave-energy 
is diminished, and the pulse is said to be " attenuated," but it 
continues otherwise uninjured and arrives " undistorted/' that 
is, with all its features intact and at the same speed as before, 
but on a reduced scale in point of size. 

This is the case of Mr. Heaviside's "distortionless circuit" 
spoken of above, and its practical realization in cables, though 
it would not at once mean Atlantic telephony, would mean 
greatly improved signalling, and probably telephony through 
shorter cables. In a cable the length of the Atlantic the 
attenuation would be excessive, unless the absence of distortion 
were secured by increasing rather the wire- conductance than 
the dielectric leakage ; but, unless excessive, simple attenua- 

* There is this difference, that in the real case the heat of dissipation 
appears locally in the wire, whereas in the imaginary case it appears 
throughout the magnetically conducting medium ; but I apprehend that 
in the imaginary case the lines would still shrink, by reason of molecular 
loops being pinched off them. 

Phil. Mag. S. 5. Vol. 47. No. 287. April 1899. 2 E 

398 Dr. Oliver Lodge on Opacity. 

tion does no serious harm. Articulation depends on the 
features of the wave, and the preservation of the features 
demands, by Fourier's analysis, the transmission of every 
frequency at the same rate. 

But now suppose any cause diminishes one of the two 
fields without diminishing the other : for instance, let the 
electric field be weakened by leakage alone, or let the mag- 
netic field be weakened by wire-resistance alone, then what 
happens ? The preservation of E and the diminution of H, 
to take the latter — the ordinary — case, may be regarded as 
a superposition on the advancing wave of a gradually growing 
reverse field of intensity SH ; and, by the relation E = /LtuH, 
this reversed field, for whatever it is worth, must mean a 
gradually growing wave travelling in the reverse direction. 

The ordinary wave is now no longer left alone and un- 
injured, it has superposed upon itself a more or less strong 
reflected wave, a reflected wave which constantly increases 
in intensity as the distance along the cable, or the penetration 
of the wave into a conducting medium, increases ; all the 
elementary reflected waves get mixed up by re-reflexion in 
the rear, constituting what Mr. Heaviside calls a diffusive 
"tail" ; and this accumulation of reflected waves it is which 
constitutes what is known as " distortion " in cables, and 
what is known as "opacity" inside conducting dielectrics. 

There is another kind of opacity, a kind due to hetero- 
geneousness, not connected with conductivity h\x' v due merely 
to a change in the constants K and //,, — properly a kind of 
translucency, a scattering but not a dissipation of energy, — 
like the opacity of foam or ground glass. 

This kind of opacity is an affair of boundaries and not of 
the medium itself, but after all, as we now see, it has features 
by no means altogether dissimilar to the truer kind of opacity. 
Conducting opacity is due to reflexion, translucent opacity is 
due to reflexion, — to irregular reflexion as it is called, but of 
course there is nothing irregular about the reflexion, it is 
only the distribution of boundaries which is complicated, the 
reflexion is as simple as ever ; — except, indeed, to some 
extent when the size of the scattering particles has to be 
taken into account and the blue of the sky emerges. But 
my point is that this kind of opacity also is after all of the 
reflexion kind, and the gradual destruction of the advancing 
Avave — whether it be by dust in the air or, as Lord Bayleigh 
now suggests, perhaps by the discrete molecules themselves, 
by the same molecular property as causes refraction and dis- 
persion — must result in a minute distortion and a mode of 
wave propagation not wholly different from cable-signalling 

Dr. Oliver Lodge on Opacity. 399 

or from the transmission of lio-ht through conductors. So that 
the red of the sunset sky and the green of gold-leaf may not be 
after all very different ; nor is the arrival-curve of a telegraph 
signal a wholly distinct phenomenon. 

There is a third kind of opacity, that of lampblack, where 
the molecules appear to take up the energy direct, converting 
it into their own motion, that is into heat, and where there 
appears to be little or nothing of the nature of reflexion. I 
am not prepared to discuss that kind at present. 

It is interesting to note that in the most resisting and 
capacious cable that ever was made, where all the features of 
every wave arrive as obliterated as if one were trying to 
sigual by heat-pulses through a slab, that even there the head 
of every wave travels imdistorted, with the velocity of light, 
and suffers nothing but attenuation ; for the superposed re- 
versed field is only called out by the arrival of the direct pulse, 
and never absolutely reaches the strength of the direct field. 
The attenuation may be excessive, but the signal is there 
in its right time if only we have a sensitive enough instru- 
ment to detect it ; though it would be practically useless as a 
signal in so extreme a case, being practically all tail. 

Nothing at all reaches the distant end till the light-speed- 
time has elapsed ; and the light-speed-time in a cable depends 
on the /jl and K of its insulating sheath, depends, if that is 
not simply cylindrical, on the product of its self-inductance and 
capacity per unit length ; but at the expiration of the light- 
speed-time the head of the signalling pulse arrives, and 
neither wire-resistance nor insulation-leakage, no, nor mag- 
netic-conductivity, can do anything either to retard it or to 
injure its sharpness : they can only enfeeble its strength, but 
they can do that very effectually. 

The transmitter of the pulse is self-induction in conjunction 
with capacity : the chief practical enfeebler of the pulse 
is wire-resistance in conjunction with capacity ; and before 
Atlantic telephony is possible (unless a really distortionless 
cable is forthcoming) the copper core of an ordinary cable 
will have to be made much larger. Nothing more is wanted 
in order that telephony to America may be achieved. There 
may be practical difficulties connected with the mechanical 
stiffness of a stout core and the worrying of its guttapercha 
sheath, and these difficulties may have to be lessened by aiming 
at distortionless conditions — it is well known also that for 
high frequencies a stout core must be composed of insulated 
strands unless it is hollow — but when such telephony is accom- 
plished, I hope it will be recollected that the full and complete 
principles of it and of a great deal else connected with tele- 

2E 2 

400 Dr. Oliver Lodge on Opacity. 

graphy have been elaborately and thoroughly laid down by 
Mr. Heaviside. 

There is a paragraph in Maxwell, concerning the way 
a current rises in a conductor and affects the surrounding 
space, which is by no means satisfactory : it is Art. 804. 
He takes the current as starting all along the wire, setting up 
a sheath of opposition induced currents in the surrounding 
imperfectly insulating dielectric, which gradually diffuse out- 
wards and die away, leaving at last the full inductive effect 
of the core-current to be felt at a distance. Thus there is 
supposed to be a diffusion of energy outwards from the wire, 
which he likens to the diffusion of heat. 

But, as Mr. Heaviside has shown, the true phenomenon is 
the transmission of a wave in the space surrounding the 
wire — a plane wave if the wire is perfectly conducting, 
a slightly coned wave if it resists, — a wave-front perpendicular 
to the wire and travelling along it, — a sort of beam of dark 
light with the wire as its core. 

Telegraphic signalling and optical signalling are similar ; 
but whereas the beam of the heliograph is abandoned to 
space and must go straight except for reflexion and refraction, 
the telegraphic beam can follow the sinuosities of the wire and 
be guided to its destination. 

If the medium conducts slightly it will be dissipated 
in situ; but if the wire conducts imperfectly, a minute trickle 
of energy is constantly directed inwards radially towards the 
wire core, there to be dissipated as heat. Parallel to the 
wire flows the main energy stream, but there is a small 
amount of tangential grazing and inward flow. The initial 
phenomenon does not occur in the wire, gradually to spread 
outwards, but it occurs in the surrounding medium, and a 
fraction of it gradually converges inwards. The advancing 
waves are not cylindrical but plane weaves, and though the 
diffusing waves are cylindrical they advance inwards, not 

I will quote from a letter of Mr. Heaviside's: — " The easiest 
way to make people understand is, perhaps, to start with a 
conducting dielectric with plane waves in it without wires 
[thus getting] one kind of attenuation and distortion. Then 
introduce wires of no resistance ; there is no difference except 
in the v 7 ay the lines of force distribute [enabling the wires to 
guide the plane waves]. Then introduce magnetic con- 
ductivity in the medium, [thereby getting] the other kind of 
attenuation and distortion. Transfer it to the wires, makino- 
it electrical resistance. Then abolish the first electric con- 
ductivity, and you have the usual electric telegraph," 

Dr. Oliver Lodge on Opacity, 401 

Opacity of Gold-leaf. 

Now returning to the general solution (5) let us apply it 
to calculate the opacity of gold-leaf to light. 

Take cr = 2000fJL square centim. per sec, 

p = 2irX 5 X 10 14 per sec. ; 

then the critical quantity 47r/pcrK or (7) is 
2x9 xlO 20 1800 

5xl0 14 x2000K/K K/K * 

This number is probably considerably bigger than unity 
(unless, indeed, the specific inductive capacity K/K of gold 
is immensely large, which may indeed be the case — refractive 
index 40, for instance, — only it becomes rather difficult to 
define) ; so that, approximately, 

// 27rup \ /40 x 5 x 10 u ,. n „ . n R 

« = V (-jF) = V 2000 = ^ 10 = 3 x 106 ; 

or the damping distance is 

F x x 10~ 5 centim. = j microcentimetre, 
whereas the wave-length in air is 

6 x 10~ 5 centim. = 60 microcentimetres. 

The damping distance is therefore getting nearer to the 
right order of magnitude, but the opacity is still excessive. 

A common thickness for gold-leaf is stated to be half a 
wave-length of light ; that is to say, 90 times the damping 
distance. Hence the amplitude of the light which gets 
through a half- wave thickness of gold is e~ 9o of that which 
enters ; and that is sheer opacity. 

[Maxwell's calculation in Art. 798, carried out numerically, 
makes the damping 

_ 2tt\xg 

e * x , = exp. (— >10 8 #) for gold, 

see equation (8) above ; or, for a thickness of half a wave- 
length, 10 -1000 , which is billions of billions of billions (indeed 
a number with 960 digits) times greater opacity than what 
we have here calculated, and is certainly wrong.] 

It must, however, be granted, I think, that the green light 
that emerges from gold-leaf is not properly transmitted ; it 
is light re-emitted by the gold *. The incident light, say the 

* This would be fluorescence, of course ; and Dr. Larmor argues in 
favour of a simple ordinary exponential coefficient of absorption even in 
metals. See Phil. Trans. 1894, p. 738, § 27. 

402 Dr. Oliver Lodge on Opacity. 

red, is all stopped by a thickness less than half a wave-length. 
The green light may conceivably be due to atoms vibrating 
fairly in concordance, and not calling out the conducting 
opacity of the metal. If the calculated opacity, notwith- 
standing this, is still too great, it is no use assuming a higher 
conductivity at higher frequency, for that would act the 
wrong way. What must be assumed is either some special 
molecular dispersion theory, or else greater specific resistance 
for oscillations of the frequency which get through ; nor 
must the imaginative suggestion made immediately below 
equation (13) be altogether lost sight of. 

There is, however, the possibility mentioned above that the 
relative specific inductive capacity of gold, K/K , if a 
meaning can be attached to it, may be very large, perhaps 
(though very improbably, see Drude, Wied. Ann. vol. xxxix. 
p. 481) comparable with 1800. Suppose for a moment that 
it is equal to 1800 ; then the value of the critical quantity (7) 
is 1 and the value of a is 

= 19xl0 5 , 

which reduces the calculated opacity considerably, though 
still not enough. 

In general, calling K/K = r, and writing the critical 

number — — as h/c. we have 

% a 2 _ I a 2 //3 2 = XV/27T 2 = ^ (7j2 _j_ c 2) _ c . 

so that aX/'Iir ranges from s/ ^1l when h/c is big, to \h/^ c 
when h/c is small. 

Writing the critical number h/c as tan e, the general value 
of a is given by 

aX=7r \/2c(sece — 1) (14) 

This is the ratio of the wave-length in air to the damping 
distance in the material in general ; meaning by " the damp- 
ing distance " the thickness which reduces the amplitude in 
the ratio e : 1. (14) represents expression (5) ; compare 
with (120- 

Theory of a Film. 

So far nothing has been said about the limitation of the 
medium in space, or the effect of a boundary, but quite 
recently Mr. Heaviside has called my attention to a special 

Dr. Oliver Lodge on Opacity. 403 

theory, a sort of Fresnel-like theory, which he has given 
for infinitely thin films of finite conductance ; it is of remark- 
able simplicity, and may give results more in accordance 
with experiment than the theory of the universal opaque 
medium without boundary, hitherto treated : a medium in 
which really the source is immersed. 

Let a film, not so thick as gold-leaf, but as thin as the 
black spot of a soap-bubble, be interposed perpendicularly 
between source and receiver. I will quote from i Electrical 
Papers,' vol. ii. p. 385 : — " Let a plane wave ^j 1 =:/jlvH 1 
moving in a nonconducting dielectric strike flush an ex- 
ceedingly thin sheet of metal [so thin as to escape the need 
for attending to internal reflexions, or the double boundary, 
or the behaviour inside] ; letE 2 = //uH 2 be the transmitted wave 
out in the dielectric on the other side, and E 3 = — /jlvH 3 be the 
reflected wave *. 

* General Principles. — It may be convenient to explain here the 
principles on which Mr. Heaviside arrives at his remarkably neat 
expression for a wave-front in an insulating medium, 

E = pvH, 
or as it may be more fully and vectorially written, 

VOE) = ^H, 
where E is a vector representing the electric intensity (proportional to 
the electric displacement), H is the magnetic intensity, and v is unit 
normal to the wave-front. E and H are perpendicular vectors in the 
same plane, i. e. in the same phase, and E H v are all at right angles to 
each other. 

The general electromagnetic equations in an insulating medium are 
perhaps sufficiently well known to be, on Mr. Heaviside's system, 

curl K = KE and —curl E = ^H, 

where " curl " is the vector part of the operator v, and where Maxwell's 
vector-potential and other complexities have been dispensed with. 

[In case these equations are not familiar to students I interpolate a 
parenthetical explanation which may be utilised or skipped at pleasure. 

The orthodox definition of Maxwell's name " curl " is that b is called 
the curl of a when the surface-integral of ft through an area is equal to the 
line-integral of a round its boundary, a being a vector or a component 
of a vector agreeing everywhere with the boundary in direction, and b 
being a vector or component of vector everywhere normal to the area. 
Thus it is an operator appropriate to a pair of looped or interlocked 
circuits, such as the electric and the magnetic circuits alwa}-s are. The 
first of the above fundamental equations represents the fact of electro- 
magnetism, specially as caused by displacement currents in an insulato?', 
the second represents the fact of magneto-electricity, Faraday's magneto- 
electric induction, in any medium. Taking the second first, it states the 
fundamental law that the induced EMF in a boundary equals the rate of 
change in the lines of force passing through it ; since the EMF or step 
of potential all round a contour is the line-integral of the electric intensity 
E round it, so that 

EMF = f Eds = - ^=-fTBe?S=-ff«HrfS : 

J cycle at J J JJ 

404 Dr. Oliver Lodge on Opacity. 

" At the sheet we have 

Ei + E 3 = E 2 
H 1 +H 3 =H 2 +47r^E 2 , 
k being the conductivity of the sheet of thickness z. Therefore 

Eg __ H 2 Ej -f E 3 __ 1 

E x H! - E7~ s= l+2«/i*8w' ' 


wherefore — ( t«H equals the curl of E. (The statement of this second 
circuital law is entirely due to Mr. Heaviside ; it is now largely adopted 
and greatly simplifies Maxwell's treatment, abolishing the * need for 
vector potential.) 

The first of the above two fundamental equations, on the other hand, 
depends on the fact that a current round a contour excites lines of mag- 
netic force through the area bounded by it, and states the law that the 
total magnetomotive force, or line-integral of the magnetic intensity 
round the boundary, is equal to 4n- times the total current through it j 
the total current being the " ampere-turns " of the practical Engineer. 

Expressing this law in terms of current density c, we write 

MMF = f Hds = 4ttC = ff 4*cd$ 

J cycle J-.' 

so always current-density represents the curl of the magnetic field due 
to it, or curl H=47rc. 

Now in a conductor c = kE, but in an insulator c = D, the rate of change 
of displacement or Maxwell's " displacement-current " ; and the dis- 
placement itself is proportional to the intensity of the electric field, 

D = — E ; hence the value of current density in general is 




whence in general 

curl H = 47r&E+KE = (4ttA'+K/?)E, 

and in an insulator the conductivity k is nothing. 

The connexion between " curl " so defined and Vv is explained as 
follows. The operator v applied to a vector R whose components are 
X Y Z gives 

. d , .. d , 7 . d 

which, worked out, yields two parts 

Q /dX,dY.dZ\ 

also called convergence, and 

T7 ./ dZ dY\ , ■ /dX dZ \ , , laY dX\ 

or say «| -\-jt] + Jc£, where £ rj £ are the components of a spin-like 
vector <». Now a theorem of Sir George Stokes shows that the normal 
component of co integrated over any area is equal to the tangential com- 
ponent of R integrated all round its boundary ; hence Vv and curl are 
the same thing. 

(•a +>s +*£)(*+.*+»>. 

Dr. Oliver Lodge on Opacity, 405 

H is reflected positively and E negatively. A perfectly con- 
ducting barrier is a perfect reflector ; it doubles the magnetic 
force and destroys the electric force on the side containing 
the incident wave, and transmits nothing/' 

[I must here interpolate a remark to the effect that though 
it can hardly be doubted that the above boundary conditions 
(tangential continuity of both E and H) are correct, yet 
in general we cannot avoid some form of sether-theory 

Whenever ox is zero it follows that R has no circulation but is 
the derivative of an ordinary single-valued potential function, whose 
dV =X.dx-{- Ydy-\-Zdz. In electromagnetism this condition is by no means 
satisfied. E and H or H and E are both full of circulation, and their 
circuits are interlaced. Fluctuation in E by giving rise to current causes 
H ; fluctuation in H causes induced E.] 

Now differentiating only in a direction normal to a plane wave 
advancing along a; the operator Vv becomes simply idjdx when applied 
to any vector in the wave-front, the scalar part of v being nothing. 

So the second of the above fundamental equations can be written 

■ dE_ dH 

~ l d^~^W 


so, ignoring any superposed constant fields of no radiation interest, E and 
II are vectors in the same phase at right angles to each other, and their 
tensors are given by E = pvYL. 

Similarly of course the other equation furnishes H=KvE; thus giving 
the ordinary K^v 2 = l, and likewise the fact that the electric and mag- 
netic energies per unit volume are equal, |KE 2 =^H 2 . 

A wave travelling in the opposite direction will be indicated by 
E= -pvH; hence, as is well known, if either the electric or the 
magnetic disturbance is reversed in sign the direction of advance is 
reversed too. 

(The readiest way to justify the equation E=^wH, a posteriori, is to 
assume the two well-known facts obtained above, viz. that the electric 
and magnetic energies are equal in a true advancing wave, and that 
0=1/ >J pK; then it follows at once.) 

Treatment of an insulating boundary. — At the boundary of a different 
medium without conductivity the tangential continuity of E and of H 
across the boundary gives us the equations 

E, + E 3 = E 2 
H 1 + H 3 =H 2 , 

where the suffix 1 refers to incident, the suflix 2 to transmitted, and 
the suffix 3 to reflected waves. 

H^-r-Hg may be replaced by /»v(Ej - E 3 ), since the reflected wave is 
reversed ; so we shall have, for the second" of the continuity equations, 

^E. 2 =mnE 2 ; 

406 Dr. Oliver Lodge on Opacity. 

when we have to lay down continuity conditions, and, 
according to the particular kind of aether-theory adopted so 
will the boundary conditions differ. My present object is to 
awaken a more general interest in the subject and to repre- 
sent Mr. Heaviside's treatment of a simple case ; but it must 
be understood that the continuity conditions appropriate to 
oblique incidence have been treated by other great mathe- 
matical physicists, notably by Drude, J. J. Thomson, and 
Larmor, also by Lord Rayleigh, and it would greatly 
enlarge the scope of this Address if I were to try to discusss 
the difficult and sometimes controversial questions which 
arise. I must be content to refer readers interested to the 
writings of the Physicists quoted — especially I may refer to 
J. J. Thomson's ' Recent Researches,' Arts. 352 to 409, and 
to Larmor, Phil. Trans, 1895, vol. 186, Art, 30, and other 

Now apply this to an example. Take k for gold, as we 
have done before, to be 1/2000 fi seconds per square centim. 
and v = Sx 10 10 centim. per sec, for v is the velocity in the 

n being the index of refraction, and m the relative inductivity. Hence, 
adding and subtracting, 

E_ 2= 2 
E, 1-hW 

E 3 \—nm m 

E,~~ 1+nm ' 
well-known optical expressions for the transmitted and reflected ampli- 
tudes at perpendicular incidence, except that the possible magnetic property 
of a transparent medium is usually overlooked. 

Treatment of a conducting boundary. — But now, if the medium on the 
other side of the boundary is a conductor instead of a dielectric, a term 
in one of the general equations must be modified; and, instead of 
curl H=KjpE, Ave shall have, as the fundamental equation inside the 

— — _ = 4ttA;L; 

or more generally (4ttZ;+K//)E. 

So, on the far side of a thin slice of thickness z, the magnetic intensity 
H 2 is not equal to the intensity H^+Hg on the near side, but is less by 

f/H = 4:TrkEdx = 4:7rkE 2 z = 4nkftvzH. a ; 
and this explains the second of the continuity equations immediately 
following in the text. 

In a quite general case, where all the possibilities of conductivity and 

capacity &c. are introduced at once, the ratio of E/H is not pv or O/K)*? 

but is ( y g-^{4iiTk-\-\i.p)~ 2 for waves in a general material medium, 

{g may always be put zero), or (R-j-pL)*(Q.+^S) 2 for waves guided by 
a resisting wire through a leaky dielectric. 

The addition of dielectric capacity to conductivity in a film is there- 
fore simple enough and results in an equation quoted in the text below. 

Dr. Oliver Lodge on Opacity. 407 

dielectric not in the conductor ; then take a film whose 
thickness z is one twenty-fifth of a wave-length of the 
incident light ; and the ratio of the transmitted to the 
incident amplitude comes out 

1/9 T, 1000 ] 

irvz 200 

Some measurements made by W. Wien at Berlin in 1888 
(Wied. Ann. vol. xxxv.), with a bunsen-burner as source of 
radiation, give as the actual proportion of the transmitted 
to the long-wave incident light, for gold whose thickness is 
10 -5 centim., '0033 or 1/300 ; while for gold one quarter as 
thick the proportion was 0'4 (see Appendix II. page 414). 

He tried also two intermediate thicknesses, and though 
approximately the opacity increases with the square of the 
thickness, it really seems to increase more rapidly : as no 
doubt it ought, as the boundaries separate. However, for 
a thickness X/25 I suppose we may assume that about l/3rd 
of the light would be transmitted, whereas the film-theory 

Simple treatment of the E.M. theory of light. — It is tempting to show 
bow rapidly the two fundamental electromagnetic equations, in Mr. Heavi- 
side's form, lead to the electromagnetic theory of light, if we attend 
specially to the direction normal to the plane of the two perpendicular 
vectors E and H, to the direction along say x, so that v = id/dx and 
v 2 = -d-jdx 2 . 

In an insulating medium the equations are 

curlH = KE and -curlE=^H; 
now curl=Vv = Vj since Sv = in this case, so 

V 2 H=KvE=KcurlE= -E>H; 
or, in ordinary form, 

<£H_ K . dm 

dx 2 ** dt 2 * 
and there are the waves. 

If this is not rigorous, there is no difficulty in finding it done properly 
in other places. I believe it to be desirable to realize things simply as 

In a conducting medium the fundamental equations are, one of them, 

curl H = K E + 4tt&E = (Kp + ink) E, 

while the other remains unchanged; unless we like to introduce the non- 
existent auxiliary g, which would make it 

and would cover wires too. 

So - v 2 H= (4tt&+I^)(#+^)H, 

the general wave equation. In all these equations p stands for djdt ; but, 
for the special case of simply harmonic disturbance of frequency pl'2ir, of 
course ip can be substituted. 

408 Dr. Oliver Lodge on Opacity. 

gives (1/200) 2 ; so even now a metal calculates out too opaque, 
though it is rather less hopelessly discrepant than it used to 
be. The result, we see, for the infinitely thin film, is inde- 
pendent of the frequency. 

Specific inductive capacity has not been taken into account 
in the metal, but if it is it does not improve matters. It 
does not make much difference, unless very large, but what 
difference it does make is in the direction of increasing 
opacity. In a letter to me Mr. Heaviside gives for the 
opacity of a film of highly conducting dielectric 

^=£ L (l + 27r f JLkvz)* + (imcz I j/v) 2 } , . . (16) 

where I have replaced his %fivzK.p last term by an expression 
with the merely relative numbers K/K and /j,//j, , called c 
and m respectively, thus making it easier to realise the 
magnitude of the term, or to calculate it numerically. 

Theory of a Slab. 
An ordinary piece of gold-leaf, however, cannot properly 
be treated as an infinitely thin film ; it must be treated as a 
slab, and reflexions at its boundaries must be attended to. 
Take a slab between x = and x=l. The equations to be 
satisfied inside it are the simplified forms of the general 
fundamental ones 

-£-»*; . -§ ME < 

k being l/<r, and K being ignored ; while outside, at # = Z, 
the condition E=yLtvH has to be satisfied, in order that a 
wave may emerge. 

The following solutions do all this if q 2 = ^irfxkp : — 

V qv —p J 

p \ qv—p J 

Conditions for the continuity of both E and H at x = 
suffice to determine A, namely if E x Hj is the incident and 
E 3 H 3 the reflected wave on the entering side, while E H 
are the values just inside, obtained by putting <r = in the 

Ei + E 3 = E , 

E 1 -E 3 =^(H 1 + H 3 )=^H . 
Adding, we get a value for A in terms of the incident light E l5 

2E 1 p(qv-p) = A{(qv+pye 2ql -(qv-p) 2 }. 

Dr. Oliver Lodge on Opacity. 409 

whence we can w T rite E anywhere in the slab, 

1 - Mw—p) f eq , . Q±±£ e 2 q i e - q A . 

E i " (qv+py^-iqv-p)* \ * gv-p J 

Put x = l, and call the emergent light E 2 ; then 

% ~ {qv+pye« l -(gv-p) 2 e-* 1 " Pi ^ ' l } 

and this constitutes the measure of the opacity of a slab, 
p z being the proportion of incident light transmitted. 

It is not a simple expression, because of course p signifies 
the operator d/dt, and though it becomes simply ip for a 
simply harmonic disturbance, yet that leaves q complex. 
However, Mr. Heaviside has worked out a complete expres- 
sion for p 2 , which is too long to quote (he will no doubt be 
publishing the whole thing himself before long), but for slabs 
of considerable opacity, in which therefore multiple re- 
flexions may be neglected, the only important term is 




?= l + (l+p/*vf ••••• ( 18 ) 

* = 4/(2wpp/a) =3xlO G 
for light in gold ; and 

V 2-7T 2-7T 1 . 

± — — — = -7777 — e = rrr about. 
av a\ 60 x 5 25 

So the effect of attending to reflexion at the walls of the slab 
is to still further diminish the amplitude that gets through, 
below the e~ al appropriate to the unbounded medium, in the 

ratio of rtg , or about a ninth. 
25 ' 

Effect of each Boundary, 

It is interesting to apply Mr. Heaviside's theory to a study 
of what happens at the first boundary alone, independent of 
subsequent damping. 

Inside the metal, by the two fundamental equations, we 

and by continuity across the boundary 

E 1 + E 3 =E , 

E 1 -E 3 =„H =At ,E (^)WJ'E , 
where still q 2 = A7rfjbkp. 

410 Dr. Oliver Lodge on Opacity. 

Therefore,, for the transmitted amplitude 
E _ 2p 

and for the reflected 

Ej p -f gv 

5_3 ^ p — qv 

B x p + qv 

or rationalising and writing amplitudes only, and under- 
standing by p no longer d/dt in general, but 'only 2tt times 
the frequency, 

Eo _ 2p 

E, ^((p + av y + M-P^' ■ ( u ) 

Any thickness of metal multiplies this by the factor e~ ax , 
and then comes the second boundary, which, according to 
what has been done above, has a comparatively small but 
peculiar effect ; for it ought to change the amplitude from p 1 
into p, that is to give an emergent amplitude 

_4/ 2. p/*v_ 

l + (l+p/avy 

instead of the above incident on the second boundary 

^ e,«-i. am 

v(i+(i+^M 3 ) ' " ' (0) 

that is for the case of light in gold, for which p/av is small, 
to change 2/ s/2 into 2 n/2, in other words, to double it. 

2 /2tt 
The effect of the first boundary alone, p 1} is — - — , or say 

1/18, and this is a greater reduction effect than that reckoned 
above for the two boundaries together. 

Thus the obstructive effect of the two boundaries together 
comes out less than that of the first boundary alone — an 
apparently paradoxical result. About one-eighteenth of the 
light-amplitude gets through the first boundary, but about 
one-ninth gets through the whole slab (ignoring the geo- 
metrically progressive decrease due to the thickness, that is 
ignoring e~ al , and attending to the effect of the boundaries 
alone ; which, however, cannot physically be done). At 
first sight this was a preposterous and ludicrous result. The 
second or outgoing boundary ejects from the medium nearly 
double the amplitude falling upon it from inside the con- 
ductor ! But on writing this, in substance, to Mr. Heaviside 
he sent all the needful answer by next post. "The incident 
disturbance inside is not the whole disturbance inside." 

Dr. Oliver Lodge on Opacity. 411 

That explains the whole paradox — there is the reflected 
beam to be considered too. At the entering boundary the 
incident and reflected amplitudes are in opposite phase, and 
nearly equal, and their algebraic sum, which is transmitted, 
is small. At the emerging boundary the incident and re- 
flected amplitudes are in the same phase, and nearly equal, 
and their algebraic sum, which is transmitted, is large — is 
nearly double either of them. But it is a curious action : — 
either more light is pushed out from the limiting boundary 
of a conductor than reaches it inside, or else, 1 suppose, 
the Telocity of light inside the metal must be greater 
than it is outside, a result not contradicted by Kundt's 
refraction experiments, and suggested by most optical 
theories. It is worth writing out the slab theory a little 
more fully, to make sure there is no mistake, though the 
whole truth of the behaviour of bodies to light can hardly be 
reached without a comprehensive molecular dispersion theory. 
I do not think Mr. Heaviside has published his slab theory 
anywhere yet. A slab theory is worked out by Prof. J. J. 
Thomson in Proc. Roy. Soc. vol. xlv., but it has partly 
for its object the discrimination between Maxwell's and other 
rival theories, so it is not very simple. Lord Kelvin's Balti- 
more lectures probably contain a treatment of the matter. 
All that I am doing, or think it necessary to do in an 
Address, is to put in palatable form matter already to a few 
leaders likely to be more or less known : in some cases 
perhaps both known and objected to. 

The optical fractions of Sir George Stokes, commonly 
written h c e f, are defined, as everyone knows, as follows. 
A ray falling upon a denser body with 
incident amplitude 1 yields a reflected 
amplitude h and a transmitted c. A 
ray falling upon the boundary of a 
rare body with incident amplitude 1 
has an internally reflected amplitude e 
and an emergent /. General prin- 
ciples of reversibility show that 
b-\-e=0, and that b 2 + cf= 1 in a trans- 
parent medium. 

Now in our present case we are attending to perpendicular 
incidence only, and we are treating of a conducting slab; 
indeed, we propose to consider the obstructive power of the 
material of the slab so great that we need not suppose that 
any appreciable fraction of light reflected at the second surface 
returns to complicate matters at the first surface. This limita- 
tion by no means holds in Mr. Heaviside's complete theory, of 
course, but 1 am taking a simple case. 

412 Dr. Oliver Lodge on Opacity. 

The characteristic number which governs the phenomenon 

is — or — -, a number which for light and gold we reckoned as 

being about gL tnat is decidedly smaller than unity, a being 

s /(2wfMkp) or a / ( - 77 ^- ). The characteristic number p/uv 

we will for brevity write as h, and we will express amplitudes 
for perpendicular incidence only, as follows : — 

Incident amplitude 1, 

externally reflected b= — < -z — \-. r^ \ 

i 1 + (I -h/i) J 


Incident again 1, 

internally reflected e— < — * i ' 

emergent /= {1 + (1 + fe)2 ^ 

(It must be remembered that e and / refer to the second 
boundary alone, in accordance with the above diagram.) 

Thus the amplitude transmitted by the whole slab, or 
rather by both surfaces together, ignoring the opacity of its 
material for a moment, is 

transmitted cf= ., — / ., ,* 7 NO « 
1 + ( 1 + A) 2 

To replace in this the effect of the opaque material, of 
thickness I, we have only to multiply by the appropriate ex- 
ponential damper, so that the amplitude ultimately trans- 
mitted by the slab is 

4;\/2.p/uv _ al 
l+(l+p/*v)* e a 

times the amplitude originally incident on its front face. 

This agrees with the expression (18) specifically obtained 
above for this case, but, once more I repeat, multiple reflexions 
have for simplicity been here ignored, and the medium has 
been taken as highly conducting or very opaque. 

But even so the result is interesting, especially the result 
for /. To emphasize matters, we may take the extreme case 
when the medium is so opaque that h is nearly zero ; then b 
is nearly — 1, c is nearly 0, being hs/2, e is the same as h 
except for sign, and /is nearly 2. 

Dr. Oliver Lodge on Opacity. 413 

An opaque slab transmits Sh 2 e~ 2al of the incident light 
energy ; its first boundary transmits only 2nh 2 . The second 
or emergent boundary doubles the amplitude. Taken in 
connexion with the facts of selective absorption and the 
timing of molecules to vibrations of certain frequency, I 
think that this fact can hardly be without influence on the 
green transparency of gold-leaf. 

Appendix I. 

Mr. Heaviside's Note on Electrical Waves in Sea- Water. 

[Contributed to a discussion at the Physical Society in June 1897 : 
see Mr. Whitehead's paper, Phil. Mag. August 1897.] 

" To find the attenuation suffered by electrical waves through 
the conductance of sea-water, the first thing is to ascertain 
whether, at the frequency proposed, the conductance is paramount, 
or the permittance, or whether both must be counted. 

"It is not necessary to investigate the problem for any particular 
form of circuit from which the waves proceed. The attenuating 
factor for plane waves, due to Maxwell, is sufficient. If its validity 
be questioned for circuits in general, then it is enough to take the 
case of a simply-periodic point source in a conducting dielectric 
(' Electrical Papers,' vol. ii. p. 422, § 29). The attenuating constant 
is the same, viz. (equation (199) loc cit.) : — 

where n/2-ir is the frequency, Tc the conductivity, c the permittivity, 
and v=(fxc)~i, fi being the inductivity. 

" The attenuator is then e~ n \ r at distance r from the source, as 
in plane waves, disregarding variations due to natural spreading. 
It is thus proved for any circuit of moderate size compared with 
the wave-length, from which simply periodic waves spread. 

" The formula must be used in general, with the best values of h 
and c procurable. But with long waves it is pretty certain that the 
conductance is sufficient to make 47rl-/cn large. Say with common- 
salt-solution ^ = (30 11 )~ 1 , then 

4ttI: _ 2kjxv 2 
en ~~ f 
if /is the frequency. This is large unless /is large, whether we 
assume the specific c/c to have the very large value 80 or the 
smaller value effectively concerned with light waves. We then 
reduce n x to 

n 1 =(2n t xJc7rf=:2r(fx7cff, 

as in a pure conductor. 

" This is practically true perhaps even with Hertzian waves, of 
which the attenuation has been measured in common- salt-solution 
by P. Zeeman. If then I— ] =30 n [and if the frequency is 300 per 
second] we get n x = about -^nnr 

Phil. Mag. S. 5. Vol. 47. No. 287. April 1899. 2 F 



Oliver Lodge on 


" Therefore 50 metres is the distance in which the attenuation 
due to conductivity is in the ratio 2*718 to 1, and there is no reason 
why the conductivity of sea-water should interfere, if the vahie is 
like that assumed above. 

"These formulae and results were communicated by me to 
Prof. Ayrton at the beginning of last year, he having enquired 
regarding the matter, on behalf of Mr. Evershed I believe. 

" The doubtful point was the conductivity. I had no data, but 
took the above I' from a paper which had just reached me from 
3Ir. Zeeman. Xow Mr. "Whitehead uses fr- 1 = 20 10 , which is no 
less than 15 times as great. I presume there is good authority for 
this datum *. Xone is given. Using it we obtain » 1 = 13 1 16 . 

" Thus 50 metres is reduced to 13-16 metres. But a considerably 
greater conductivity is required before it can be accepted that the 
statements which have appeared in the press, that the failure of 
the experiments endeavouring to establish telegraphic communica- 
tion with a light-ship from the sea-bottom was due to the con- 
ductance of the sea, are correct. It seems unlikely theoretically. 
and Mr. Stevenson has contradicted it (in 'Nature') from the 
practical point of view. So far as I know, no account has been. 
published of these experiments, therefore there is no means ol 
finding the cause of the failure." 

Appendix II. 
The experiments of W. Wien on the transparency of metals, 
bv means of a bolometer arranged to receive the radiation from a 
bunsen burner transmitted through different films, resulted in the 
following numbers for the proportion of radiation transmitted. 

Proportion transmitted. 


Thickness in 
10~ : centim. 

Bunsen burner 

Bunsen burner 









Iron & Platinum 





Gold 1 











Gold 3 

Gold 4 

Silver 1 (blue) ... 

Silver 2 (grey) . . . 





Silver 3 (grey) . . . 





Silver 4 (blue) ... 





Silver 5 (grey) ... 





* Dr. J. L. Howard has recently set a student to determine the resistivity 
of the sea-water used by Professor Herdman, density 1-019 gr. per c.c, 
and he finds it to be 3x 10 10 c.g.s. at 15° C— O. J. L*, March 1899. 

Prof. J. J. Thomson on ilie Cathode Rays. 415 

The thickness is in millionths of a millimetre, i. e. is in terms 
of the milli-microm called by microscopists p/i. 

The films were on glass, and the absorption of the glass was 
allowed for by control experiments. 

It is to be understood that of the whole incident light the pro- 
portion reflected is first subtracted, and the residue is then called 
1 in order to reckon the fraction transmitted of that which enters 
the metal, it being understood that the residue which is not trans- 
mitted (say *68 or *63 in the case of platinum) is absorbed. It 
may be that more and better work has been done on the opacity of 
metals than this : at any rate there seems to me room for it. I 
do not quote these figures with a strong feeling of confidence in their 
accuracy. They are to be found in Wied. Ann. vol. xxxv. p. 57. 

XXXVI. Note on Mr. Sutherland's Paper on the Cathode Rays. 

To the Editors of the Philosophical Magazine. 

IN the March number of the Philosophical Magazine 
Mr. Sutherland considers a theory of the cathode rays- 
which I published in this Journal in October 1897, and in 
which the carriers of the charges were supposed to be the 
small corpuscles of which the atoms of the elementary bodies 
could, on an extension of Prout's hypothesis, be supposed to 
be built up. Mr. Sutherland takes the view that in the 
cathode rays we have disembodied electric charges, charges 
without matter — electrons — their apparent mass being due to 
the energy due to the magnetic force in the field around them 
varying as the square of the velocity (see Phil. Mag. xl. p. 229). 
I may say that the view that in the cathode rays the con- 
stancy of the mass arose from the charge being torn away 
from the atom, so that we had only the effective mass due to 
charge, occurred to me early in my experiments, but except 
in the form (which I gather Mr. Sutherland does not adopt), 
and which only differs verbally from the view I took, that the 
atoms are themselves a collection of electrons, that is, consti- 
tute an assemblage of particles the individuals of which are 
the same as the carriers in the cathode rays ; this conception 
seemed to me to be wanting in clearness and precision, and 
beset with difficulties from which the other was free. In 
the theory which I gather Mr. Sutherland holds of the 
cathode rays, we have atoms which are comparatively large 
systems ; these can be charged with electricity, of which in 
electrons and neutrons we have what correspond to atoms 
and molecules, the radius of an electron being about 10~ u cm. 
What conception must we form of the connexion between the 

416 Prof. J. J. Thomson on the Cathode Rays. 

above and the electron when the atom is charged? The charged 
atom cannot behave as if the charge were spread over its 
surface ; for if it did it would require a potential fall of about 
a million volts to separate the electron from the atom. Again, 
the value of m/e as determined by the Zeeman effect is of the 
samo order as that deduced from the deflexion of the cathode 
rays, so that the charge must move independently of the body 
charged. The electron thus appears to act as a satellite to 
the atom. 

A difficulty in the way of supposing that mass is entirely 
an electrical effect, and that in the impact of cathode rays we 
have electrons striking against much larger masses, is the 
large proportion of the energy converted into heat when the 
cathode rays strike against a solid. AVhen an electron is 
stopped, theory shows that the energy travels off in a pulse of 
electromagnetic disturbance, and this energy would only 
appear as heat at the place struck if the waves were absorbed 
by the target close to the point of impact : if these targets 
were made of a substance like aluminium, which is trans- 
parent to these waves, we should expect much of the energy 
to escape in the pulse. As far as I can see the only ad- 
vantage of the electron view is that it avoids the necessity of 
supposing the atoms to be split up : it has the disadvantage 
that to explain any property of the cathode rays such as 
Lenard's law of absorption, which follows directly from the 
other view, hypothesis after hypothesis has to be made : it 
supposes that a charge of electricity can exist apart from 
matter, of which there is as little direct evidence as of the 
divisibility of the atom ; and it leads to the view that cathode 
rays can be produced without the interposition of matter at 
ail by splitting up neutrons into electrons : it has no ad- 
vantage over the other view in explaining the penetration of 
solids by the rays, this on both views is due to the smallness 
of the particles. Until we know something about the vibra- 
tions of electrons, it does not seem to throw much light on 
Eontgen rays to say that these are vibrations of the electrons. 
The direct experimental investigation of the chemical 
nature (so to speak) of the cathode rays is very difficult, and 
though I have for some time past been engaged on experi- 
ments with this object, they have not so far given any decisive 

Yours very sincerely, 

J. J. Thomson. 
Cavendish Laboratory Cambridge, 
March 11th, 1899. 

[ 417 ] 

XXXVII. Notices respecting New Books. 

Harper's Scientific Memoirs. Edited by Dr. J. 8. Ames, 
Professor of Physics in Johns Hopkins University, Baltimore. — 
I. The Free Expansion of Gases ; Memoirs by Gray-Lussae, Joule, 
and Joule and Thomson. — II. Prismatic and, Diffraction Spectra ; 
Memoirs bv J. you Fraunhofer. New York & London : Harper 
& Bros., 1898. 

r PHESE two volumes form the commencement of a series of 
-■- memoirs on different branches of physics, each containing the 
more important epoch-making papers in connexion with the 
subject of the memoir. Professor Ames, in addition to editing the 
series, contributes the translations of the papers by Gruy-Lussac 
and Fraunhofer in the first and second volume respectively, and 
in subsequent volumes such subjects as " The Second Law of 
Thermodynamics," " Solutions," " The Laws of Gases," and 
" Kontgen Kays " will, among others, receive similar treatment. 
Each paper will be enriched by notes aud references, and the 
bibliography of the subject will be given in an appendix to each 

The series will serve to bring before English-speaking readers 
the principal foreign classical papers on physical subjects, and 
the reprinting of the papers published in the numerous and 
frequently inaccessible journals issued in this country should prove 
a great convenience. The list of American physicists who have 
undertaken a share of the editing is a guarantee that the work will 
be done with the characteristic industry of our friends across the 
Atlantic. J. L. H. 

XXXVIII. Proceedings of Learned Societies. 


[Continued from p. 328.] 

December 7th (cont.) — W. Whitaker, B.A., F.R.S., President, 
in the Chair. 

2. 'The Permian Conglomerates of the Lower Severn Basin.' 
By W. Wickham King, Esq., F.G.S. 

The rocks thus described are the calcareous conglomerates in- 
cluded in the Middle Permian of the Shropshire type, and exposed 
north of the Abberley and Lickey Hills. Three calcareous horizons 
occur, interstratified in sandstones or marls and surmounted by 
the Permian breccia. It was the opinion of Ramsay and others 
that the materials of the calcareous horizons and of the Permian 
breccia had been brought from the Welsh border ; but Buckland and 

418 Geological Society : — 

Jukes, among others, claimed a southern derivation for those of the 
Permian breccia, from local hill-ranges to the south. The latter 
view accords with the fact that the pebbles composing these 
calcareous horizons, and also the broken fragments constituting the 
Permian breccias north of the Abberley and Lickey Hills, are coarser 
in the south-easterly direction, and gradually become finer to the 

The fragments embedded in the Middle Permian calcareous bands 
near the Lickey are chiefly of Archaean rocks, but in all the other 
districts described there are very few rock-fragments older than 
Woolhope Limestone. On the other hand, pebbles of dolomitic Wen- 
lock and Carboniferous Limestones are abundant, while Aymestry 
Limestone, Old Bed, Carboniferous, and Lower Permian sandstones 
occur in greater or less abundance ; and all these rocks, except the 
Carboniferous Limestone, may be seen in situ near at hand to the 
south. A summary of work done in the Halesowen Coal-Measure 
conglomerates and in the Permian breccia north of the Abberley 
and Lickey Hills is given, to bring out one of the lines of argument 

(1) Ridges near the Lickey were denuded down to the Archaean 
rocks in Upper Carboniferous time ; therefore, as might have been 
expected, both the adjacent Upper Carboniferous conglomerate and 
the Middle Permian calcareous cornstoues are composed of such 
fragments of Archaean rocks as are to be found in situ there, or at 
Nuneaton ; and the Upper Carboniferous conglomerate is also largely 
composed of Palaeozoic rocks identical with those in situ on the 
flanks of the Lickey. 

(2) The Middle Permian calcareous conglomerates of the other 
districts described are for the most part made up of fragments not 
older than the Woolhope Limestone, which were presumably derived 
by denudation from ridges which had become more extensive. 

(3) The Lickey ridges having been denuded to the Archaean rocks 
and the more extended area to the Woolhope Limestone, the later 
Permian breccias are composed of Archaean fragments near the 
Lickey, but of rocks not newer than the Woolhope Limestone in the 
other districts north of the Abberley and Lickey Hills. 

The author has for several years called the ancient ridges from 
which these materials were derived the ' Mercian Highlands,' and 
claims that the Palaeozoic and Archaean rocks composing the stumps 
of these highlands lie almost entirely buried under the Trias of 
the Midlands south and east of the S.E. Shropshire and South 
Staffordshire regions. 

December 21st.— W. Whitaker, B.A., F.R.S., President, 
in the Chair. 

The following communications were read : — 

1. " On a Megalosauroid Jaw from Rhaetic Beds near Bridgend, 
Glamorganshire.' By E. T. Newton, Esq., F.R.S., F.G.S. 

On the Torsion- Structure of the Dolomites. 419 

2. ' The Torsion-Structure of the Dolomites.' By Maria M. 
Ogilvie, D.Sc. [Mrs. Gordon]. 

The paper opens with a geueral account of the work of Eichthofen, 
Mojsisovics, Bothpletz, Salomon, Brogger, the author, and others 
on the Dolomitic area of Southern Tyrol. It then gives the results of a 
detailed survey recently made by the author of the complicated strati- 
graphy of the rocks of the Groden Pass, the Buchenstein Valley, and 
the massives of Sella and Sett Sass ; together with the author's inter- 
pretation of these results, and her application of that interpretation 
to the explanation of the Dolomite region in general. The author 
concludes that overthrusts and faults of all types are far more common 
in the Dolomites than has hitherto been supposed. The arrangement 
of these faults is typically a torsion-phenomenon, the result of the 
superposition of a later upon an earlier strike. This later crust- 
movement was of Middle Tertiary age, and one with the movement 
which gave origin to the well-known Judicarian-Asta phenomena. 
The youngest dykes (and also the granite-masses) are of Middle 
Tertiary age, while the geographical position of both is the natural 
effect of the crust-torsion itself. This crust-torsion also fully 
explains the peculiar stratigraphical phenomena in the Dolomite 
region, such as the present isolation of the mountain-massives of 
dolomitic rock. 

The Groden Pass area, first selected for description by the author, 
is a distorted anticlinal form running approximately JNT.N.E. and 
S.S.W., and including all the formations ranging from th.e Bellerophon- 
Limestone, through the Alpine Muschelkalk and Buchenstein Beds, 
to the top of the Wengen Series. When studied in section, the strata 
of the Pass are found to be arranged in a complex fold form, showing 
a central anticlinal with lateral wings, limited on opposite sides by 
faults and flexures. Strongly marked overthrusting to S.S.E. in 
the northern wing is responded to by return overthrusts to N.N.W. 
in the southern wing. The strata in the middle limb of the anti- 
clinal wings bend steeply downwards into knee-bend flexures. 
Through these run series of normal and reversed faults, into which 
has been injected a network of igneous rocks, giving rise to ' shear- 
and-contact ' breccias, which have previously been grouped as 
Buchenstein tuff and agglomerates, and referred to the Triassic 

The area of movement of the Groden Pass system is an ellipsoid in 
form. Two foci occur within it, where the effects of shear and strain 
have culminated. The forces of compression acted not in parallel 
lines, but round the area, thus causing torsion of the earth-crust. 
Two main faults occur (with a general east-and-west trend) whose 
actual lines of direction intersect at a point about midway between 
the foci of the torsion-ellipsoid. These are the chief strike torsion- 
faults ; many minor ones pass out easterly and westerly from the foci, 
forming longitudinal or strike torsion-bundles. The strike 
system of faults is cut by a series of diagonal or transverse 
curved branching faults, with a more or less north-easterly or north- 
westerly direction. These diagonal faults may cut each other, or 

420 Geological Society. 

may combine to form characteristic torsion-curves. The author 
regards the longitudinal and diagonal faults as constituting one 
system. Each portion on one side of the anticlinal form of the 
system has its reciprocal on the other side. The Spitz Kofi syncline 
on the north is the reciprocal of that of Sella on the south, the 
Langkofl on the south-west of that of Sass Songe on the north-east, 
and so on. 

The anticlinal area of the Buchenstein Yalley is next described. 
Here we have a torsion-system similar to that of the Groden Pass, 
and made up of similar elements ; but the western portion of the 
anticlinal is much compressed and displaced. Opposing areas of 
depression are also found here, that of Sella and Sett Sass on the 
north being reciprocated by that of the Marmolata on the south, and 
soon. The porphyrite-sills have here been mainly injected into the 
knee-bends of the northern wing of the anticlinal form, but igneous 
injections and contact-phenomena are also met with in some of the 
transverse faults. 

A full description is given of the sequence and stratigraphy in the 
Sella massive — once regarded by some authorities as a Triassic 
coral-reef. This is an ellipsoidal synclinal area with X.X.E. and 
S.S.AV. axes twisted to north-east and south-west. Peripheral over- 
thrusts have taken place outward from the massive, in such a way 
as to buckle up the rocks like a broad-topped fan -structure, and these 
overthrusts are traced by the author completely round the massive. 
A central infold of Jurassic strata occurs on the plateau, where the 
Upper Trias has been overthrust inwards on three sides of the infold. 
The author next passes in review the results obtained in the area 
of Sett Sass, etc. and shows how they all present corresponding 
tectonic phenomena. 

The district thus studied in detail by the author forms a typical 
unit in the structural features of the Dolomite region. It is cut off 
to the eastward by the limiting fault (north-and-south N of Sasso 
de Stria, and to the westward by the parallel fault of Sella Joch. 
These are definite confines, which limit a four-sided area, influenced 
by the Groden Pass torsion-system on the north and the Buchenstein 
Valley system on the south. The limits of this four-sided figure 
include a compound area of depression (formed by the Sella and Sett 
Sass synclinals) traversed by the diagonal Campolungo buckle. 
4 The area displays in a marked degree the phenomena of interference 
cross-faults cutting a series of peripheral overthrusts round the 
synclines, and parallel flexure faults between the anticlinal buckles 
and the synclinal axes/ 

In conclusion, the author applies her results to the interpretation 
of the complexities of the Judicarian-Asta region of the Dolomites 
in general, aud also to the explanation of the characteristic 
structural forms of the Alpine system as a whole. 

PHI. Mag. S. 5. Vol. 47. P]. I[[ 


■ < 







[FIFTH SERIES.] |' y MAY 15 1899 


MA Y 1899. 

XXXIX. The Effects of Temperature and of Circular Mag- 
netization on Longitudinally Magnetized Iron Wire. By 
F. H. Pitcher, M.A.Sc, Demonstrator in Physics, McGill 
University, Montreal* . 

Objects of the Investigation. 

IN commencing these experiments in October 1894, the 
original intention was to investigate only the effect of 
temperature on the magnetization of iron. With this object 
the specimen was heated in a platinum tube, maintained at a 
steady temperature by means of an electric current. The 
temperature was inferred from the resistance of the platinum 
tube, which was very uniformly heated and extended for some 
distance beyond the ends of the iron wire. This proved to be 
a very perfect method of heating, as the temperature could 
be easily varied and accurately regulated and measured. The 
current in the platinum tube was also without magnetic effect 
on the specimen or the magnetometer, and the specimen was 
necessarily at the same mean temperature as the enclosing 
tube. A concentric brass tube formed the return lead. 

Unfortunately there was some difficulty at the outset in 
procuring suitable platinum tubes, and the attempt to make 
tubes in the laboratory by rolling up strips of platinum foil 
did not prove entirely satisfactory owing to the inferior 
quality of the foil. The tubes invariably cracked and became 
useless before a complete series of observations had been 

* Communicated by Prof. H. L. Calleudar, M.A., F.R.S. 
Phil. Mag. S. 5. Vol. 47. No. 288. May 1899. 2 G- 

422 Mr. F. H. Pitcher on Effects of Temperature and 

obtained. The incomplete series of tests obtained in this 
manner were, however, of interest as a verification of the 
method subsequently adopted. 

In the meantime, while awaiting the production of suitable 
tubes, it was thought that interesting results might be ob- 
tained by heating the iron wire with an electric current 
passed through the wire itself, and deducing its mean tem- 
perature from its resistance, with the aid of the formula 
verified by Prof. Callendar (Phil. Trans., A. 1887, p. 225) by 
the direct comparison of platinum and iron wires. The 
objection to this method of heating is that the wire is cir- 
cularly magnetized by the heating current, and that it is 
necessary to disentangle the effects of the temperature change 
and of the circular magnetization on the longitudinal mag- 
netization of the specimen. The effect of the circular field 
itself, however, is not without interest* 

In order to disentangle these effects, three separate series of 
observations were taken for the same range of current or circu- 
lar field, (1) in a very high vacuum ; (2) in air at atmospheric 
pressure; (3) in a current of water. In case (1) a current 
of 16 amperes sufficed to heat the wire above its critical 
temperature ; in case (2) the highest temperature was 
400° C. : in case (3) the heating effect was practically 

Methods of Measurement Adopted. 

The iron wire specimen was magnetized by means of a 
specially constructed solenoid, and the intensity of mag- 
netization I at any time was observed by means of the 
deflexion of a magnetometer, the direct effect of the solenoid 
being very carefully compensated by means of a balancing 
coil in the usual manner. The broadside- on position was 
adopted for the test in preference to the vertical or the end- 
on position, as it had been found by preliminary tests that, 
if the distance of the specimen from the magnetometer were 
suitably adjusted, the broadside-on method agreed much 
more closely with ballistic tests of the same specimen than 
either of the more usual positions. The value of the Earth's 
field H was repeatedly determined by the aid of a Kohlrausch 

The deflexion of the magnetometer was observed by means 
of a telescope and a metre-scale of milk-glass very accurately 
divided. The magnetometer was also provided with suitable 
galvanometer-coils, so that readings of current and resistance 
could be taken on the same scale. The scale of the gal- 
vanometer was carefullv calibrated throughout, and all the 

Circular Magnetization on Magnetized Iron Wire. 423 

observations were reduced by means of the correction curve 
obtained. The heating and magnetizing currents were 
passed through suitable manganin resistances immersed in 
oil, so that by observing the deflexion of the galvanometer 
when connected successively to the terminals of the manganin 
resistances and the specimen itself, the two currents and the 
resistance of the specimen could be quickly determined at any 
time with an accuracy of at least 1 part in 1000. By varying 
the resistance in series with the galvanometer it was possible 
to obtain accurately readable deflexions through a very wide 
range of current and resistance. 

The Magnetizing Solenoid. — The solenoid was wound on a 
thin brass tube about 70 centim. long, with an external 
diameter of 2*23 centim. The insulation resistance was very 
high, special pains having been taken to insulate each layer 
with paraffin and paper. The winding was tested for uni- 
formity by measuring each fifty turns during the process. 
The length of the winding was 60 - 25 centim., containing 
4079 turns of No. 24 B & S double silk-covered wire in four 
layers, and having a resistance of 28 ohms at 15° G. This 
gave a magnetizing field of over 300 C.G.S. with 100 volts 
on the terminals. In order to dissipate the heat due to the 
magnetizing current at high fields, an internal water circula- 
tion was provided through an annular space formed by a 
second concentric brass tube. 

The Mounting for the Iron Wire Specimen. — As the specimen 
was enclosed in a vacuum-tube, and its resistance at each 
temperature determined, a special form of mounting was 

The iron-wire specimen was 0'127 centim. diameter and 26*1 
centim. long, or a little over 200 diameters. Its ends were 
fused to copper wires *040 centim. diameter and 10 centim. 
long ; the diameter of the copper wire being chosen by trial 
to give a uniform temperature throughout the whole length 
of the iron wire when the heating current was passed through 
the circuit. 

The ends of these copper wires were tin-soldered and 
riveted to stout copper conductors wdiich were brought out 
through spiral copper springs to the ends of the containing 
tube. The two copper springs, whose function was to take 
up the slack of the heated specimen, had each exactly the 
same number of turns, and were wound oppositely so that the 
direct effect (on the magnetometer) of the current circulating- 
through them would be compensated. The whole was centered 
and kept in place by brass washers which fitted the thin glass 
containing tube. The glass tube just fitted the inner brass 

2 G2 

424 Mr. F. H. Pitcher on Efects of Temperature and 

tube, and was made sufficiently long to extend at both ends 
beyond the brass tube. 

Very fine platinum wires (0*003 centim. diameter) were 
attached at 15 centim. apart to the iron wire specimen. They 
served as potential leads, and were brought out beyond one 
end of the glass containing tube, through sealed capillary 

The glass tube was made tight at both ends by fitting 
brass cups over and filling with fusible alloy, one end of the 
tube having been drawn down so that when capped it could 
be slipped into the solenoid tube. A copper tube was intro- 
duced through and soldered in the larger brass cap, to serve 
for exhausting. The vacuum was maintained by a five-fall 
Sprengel pump, assisted in the early stages by a water pump. 

The remaining apparatus consisted mainly of resistance- 
boxes, rheostats, special arrangements of mercury-cup con- 
tacts, switches, storage-batteries, &c. 

Preliminary Tests. 

The specimen was of commercial so-called soft iron wire, 
and was carefully annealed and polished before mounting. 
The vacuum-tube containing it was connected to the Sprengel 
pump and a high vacuum maintained while the wire was 
being heated by the current. It was observed by the eye 
that the heating was very uniform, the whole becoming 
an even red right up to the ends, at a high vacuum. The 
zero-point or resistance at 0° C. of the specimen was now 
obtained. It was then placed in the solenoid and the 
equivalent magnetic length determined. This was found to 
be a little over 20 centim., and the magnetometer distance 
was arranged so that slight changes of the length had a 
minimum effect on the magnetometer readings. A pre- 
liminary test for the magnetic quality of the iron at ordinary 
temperatures w T as first made. It was found before further 
annealing to be fairly hard, having a hysteresis loss for 
B = 17,000, of 16,000 ergs per cub. centim., and a permeability 
at that induction of 500. 

After annealing several times in a vacuum, the loss at 
nearly the same induction had fallen to 6000 ergs, and finally, 
after successive annealings, arrived at the extraordinarily low 
value of 557 ergs for B = 3500 at ordinary temperatures. 
This, in spite of the fact that the specimen was only com- 
mercial wire, is almost as good as the best specimen of 
transformer iron tested by Ewing. By this time the wire 
had settled down to a very steady magnetic state, as shown 

Circular Magnetization on Magnetized Iron Wire. 425 

by successive tests before and after heating. Before pro- 
ceeding further the zero-point of the specimen was again 
tested, and was found to agree to within 1/10 of one percent, 
with the previous determination. There was no trace of 

The Observations. 

The method of taking the observations was as follows : — 
First, the containing tube was exhausted. The magnetometer 
deflexions were observed at longitudinal fields ranging from 
1 to 30, with currents in the wire varying from to 1 6 amp. 
It was previously observed that the wire was practically 
demagnetized at 16 amps., which corresponds to a temperature 
of 750° C. 

The current in the solenoid was reversed several times 
before each reading of the magnetometer deflexion, thus 
ensuring a reversal curve. The current in the wire was kept 
constant for each reversal curve. Its value with that of the 
corresponding resistance of the wire was observed at intervals 
along the curve. The effect of residual thermal currents in 
the heating circuit was eliminated by reversal of the current 
in the wire. 

The Sprengel pump was kept running during the whole 
set of observations in order that the gases given off from 
the heated iron and copper, as well as air which might leak 
in owing to imperfect sealing, might not affect the vacuum. 
Under these conditions the vacuum was kept very high and 
constant, and the iron wire remained bright throughout the 
whole series of tests. 

On the completion of this set of observations the vacuum 
was let down, the containing tube disconnected from the 
pump, and a similar set taken in air. All the conditions 
remained the same as before, except that the wire was tested 
in air instead of in a high vacuum, and was therefore at a 
necessarily lower temperature for the same heating current. 

As soon as possible afterwards, two similar series of obser- 
vations were taken at much higher fields, varying from 50 to 
300. The conditions were exactly the same in this case as in 
the lower fields, except that the controlling field of the 
magnetometer had to be strengthened, and that the vacuum 
at which the higher temperature observations were taken was 
slightly less perfect. 

At this stage the zero-point of the specimen was again 
tested and was found to agree with the two previous deter- 
minations, within the limits of accuracy of the method. It is 
interesting to observe that the electrical resistance was 

426 Mr. F. H. Pitcher on Effects of Temperature and 

practically unaffected by magnetization, and that the reduc- 
tion of the hysteresis loss to one-third of its original value 
was unaccompanied by any measurable change of conductivity. 
Specimen tables of the reduced observations in vacuum and 
in air are here exhibited. 

Table I. — Longitudinal Fields 1-30. 

In Vacuum. 

In Air. 

Current in Wire 12-45 amp. 
Eesistance of Wire 0-08324 w. 
Temperature of Wire 552° 0. 

Current in Wire 12*60 amp. 
Eesistance of Wire 003447 w. 
Temperature of Wire 224° C. 


















Table II.— Longitudinal Fields 50-290. 

In Vacuum. 

In Air. 

Current in Wire 12 amp. 
Eesistance of Wire 007312 w. 
Temperature of Wire 496° C. 

Current in Wire 1T60 amp. 
Eesistance of Wire 03035 w. 
Temperature of Wire 187° C. 
















Circular Magnetization on Magnetized Iron Wire. 427 

















©. — 


— . 




s < 






































































? th& 























































Ajmperes Jn t/i& w/re. 

Figs. 1 and 2. Containing Results of Direct Observation. 

428 Mr. F. H. Pitcher on Effects of Temperature and 

The Curves of fig. 1. — The magnetic observations were 
taken at thirteen different fields, curves for seven of which 
have been plotted in fig. 1. The longitudinal field of the 
solenoid is indicated for each pair of curves. Each pair is 
drawn for the same field. Abscissae represent current in 
the wire and ordinates longitudinal intensity of magnetization. 
The curves drawn in full lines are for the observations taken 
in a vacuum and at a higher temperature than those in dotted 
lines, which are for the observations taken in air. Where the 
curves coincide, showing no effect of temperature, full lines 
are drawn. 

The curves shown in fig. 2 are the corresponding tempe- 
rature-curves plotted to current in the wire. The lower of 
these curves is that for the specimen in air. The points 
marked with a cross belong to the observations in fields above 
30 c.G.s. The middle curve is the temperature-curve in 
vacuum for the higher fields ; while the top curve is that for 
lower fields in vacuum, from 30 down. The temperature- 
difference between the dotted and full-line curves for any 
current in the wire can at once be found by consulting the 
corresponding temperature-curves in fig. 2 on the same 

Considering the tables and the curves of figs. 1 and 2, it 
will be noticed that the known behaviour of soft iron at constant 
fields, as temperature advances, is well displayed. 

In high fields the dotted and full-line curves of each pair 
separate almost from the start, and do not meet in any part 
of their course, showing a continual decrease in intensity of 
magnetization from the beginning as temperature increases. 

At a field of 10 c.G.s. there appears to be no change in 
the intensity of magnetization for a temperature-difference of 
350° C, as shown by the fourth curve (fig. 1) from the top, 
together with the top and bottom curves (fig. 2) . 

At still lower fields the intensity begins to increase for a 
comparatively small temperature-difference, as shown by the 
two lower pairs of curves, fig. 1. The effect of the circular 
field on the longitudinal component is here very marked. If 
the dotted curves, where the temperature is less in evidence, 
be considered, it will be seen that, in high longitudinal fields, 
as the circular field increases there is but little change in the 
longitudinal intensity. Somewhere between a longitudinal 
field of 30 and 10 a point of inflection occurs, and the curves 
below are changed in form entirely. At fairly low fields 
(from 3*5 downwards) the drop with small increments of 
circular field is at first very great, but soon reaches a limit ; 
and the curves become very flat. 

The explanation of these effects is that for high longitudinal 

Circular Magnetization on Magnetized Iron Wire. 429 

fields the permeability is very small. Therefore the circular 
field would have at first only a slight effect in diminishing the 
longitudinal intensity. On the other hand, in lower fields 
the permeability is many times greater, and hence the effect 
of the circular field is much more marked, until the direction 
of the resultant field swings around nearer to the direction of 
the circular field, when the rate of change in the longitudinal 
intensity becomes very slow. 

Temperature- Curves, fig. 2. — The curious wave which occurs 
at the upper end of each high-temperature curve {in vacuo) 
may be partly due to the sudden change in the temperature- 
coefficient of iron at high temperatures *, and partly also, in this 
particular case, to the effect on the vacuum of gas given off 
from the wire. The pump may not have been able to exhaust 
at a sufficiently high rate. 

Hopkinson (Phil. Trans, vol. clxxx.) investigated the resist- 
ance-temperature curves of soft iron and steel at high tempe- 
ratures up to 900° C. The temperature was inferred from 
the resistance of a copper wire enclosed with his specimens, 
apparently on the assumption of a constant temperature- 
coefficient for copper. He found a sudden drop in the tem- 
perature-coefficient for soft iron and steel between 800° and 
900° C. beyond the critical point. It seems desirable that 
this should be tested up to higher temperatures by comparison 
with a platinum pyrometer. 

Method of Distinguishing the Effects of Temperature and 
of Circular Magnetization. 

By treating the ordinates of the curves in fig. 1 as one 
component of the resultant intensity the temperature-variation 
of the magnetization of iron at high fields can be worked out 
to a fairly accurate result. 

The first step in the reduction was to obtain a family of 
curves {a) of average resultant I and H at different tem- 
peratures. These were compared with a similar set (b) from 
which temperature-effect had been eliminated. Then by 
treating the drop between corresponding curves of the first 
and second set — at the same resultant fields — as due to tem- 
perature, the temperature-effect on the resultant intensity 
was obtained. 

The average circular field in the wire was taken equal to 
two thirds of the field at the periphery. This was compounded 
with the longitudinal field to give the average resultant field 
due to the two magnetizing forces. The longitudinal field 
previous to compounding was corrected for the effect of the 

* A rapid increase of a similar character was observed by Callendar to 
occur just below the critical point. 

430 Mr. F. H. Pitcher on Effects of Temperature and 

ends of the specimen. The corresponding value of the average 
resultant permeability was taken from the full-line curves 

(fig- i). 

Effect of Circular Magnetization (fig. 3). 

Before obtaining the second set (b) of average resultant I 
and H curves with which the above were compared, it was 
necessary to eliminate the effect of temperature. This was 
done by taking the drop between any two points on the dotted 
and full-line curves (fig. 1) which are at the same tempe- 
rature as due to circular magnetization, an assumption which 
is very nearly correct, especially in the higher fields. 

In this way the family of curves in fig. 3 was obtained. 


•X soc 



Effect of Circular Field at Constant Temperature 18° C. on the 
I-H Curves of Longitudinal Magnetization. 

The ordinates are longitudinal magnetization,, and the abscissae 
longitudinal field. The curves in the lower part of the figure 
are the continuation in higher fields of those above. They 
are plotted to the same scale of I, but the H scale is reduced 

Circular Magnetization on Magnetized Iron Wire. 431 

ten times. The curves are all drawn for the same tempe- 
rature, viz. 18° C, and the current in the wire is indicated in 
each case. The correction-line for the length of the specimen 
is also drawn. 

Method of Deducing the Temperature- Curves (fig. 4). 

By compounding the ordinates of the curves in fig. 3 with 
the corresponding value of the average circular intensity, the 
set of resultant I and H curves (b) at constant temperature 
for comparison with the corresponding set (a) at different 
temperatures was obtained. These two sets of curves are not 
shown in the figures. They were only a step in the reduction, 
and were not intrinsically interesting. 


= &f 


f o 











9 l 


















Temp, /n Degrees Cent". 

Effect of Temperature on the Magnetization of Iron in High Fields. 

The final result showing the effect of temperature in high 
fields is shown in fig. 4. Here ordinates represent average 
resultant intensity and abscissa? temperature in degrees centi- 
grade. Each curve is drawn for a constant average resultant 

432 Effects of Temperature on Magnetized Iron Wire. 

Verification and Discussion of Results. 

It is interesting to notice in fig. 3 the limiting effect of the 
circular magnetization in high fields. The curves up to 
1 2 amps, in the wire almost coincide at a. longitudinal field 
of 150 C.G.S. 

To test how far the results in fig. 3 were reliable, the wire 
was mounted in a glass tube and a water circulation allowed 
to flow through the tube in contact with the wire. Thus the 
temperature of the wire was kept practically constant for all 
values of the current. The results of these tests agreed very 
closely indeed with those shown in fig. 3, even down to a field 
of 10 C.G.S. 

Klemencic (Wied Ann. vol. lvi. p. 574) investigated the 
circular magnetization of iron wires together with the axial 
magnetization by a different method. By including the wire 
as an arm of a Wheatstone's bridge and using a ballistic 
galvanometer, the circular magnetization was deduced from 
the observed value of the self-induction for different currents 
in the wire. Here the change of temperature of wire intro- 
duces difficulties. It seems that the magnetometer method, 
when the wire is kept at a constant temperature, is much 
simpler and less troublesome. 

The results shown in fig. 4 were found to agree closely with 
the tests obtained for soft iron by the platinum-tube method. 
The point of demagnetization was obtained a trifle lower by 
the latter method, and the initial slope of the curve at a field 
of 290 c.G s. was a little less. It will be noted that the point 
of demagnetization is not absolutely sharp ; the curves suddenly 
change their direction and I decreases more slowly. This 
was also investigated in special tests made by the platinum- 
tube method. The value of I at 750° C. was observed to be 
about 7 C.G.S. 

These results are found to agree very well with those for 
soft iron in high fields obtained by Curie (Comptes Rendus, 
vol. cxviii. p. 859). He heated his specimen in a platinum 
heating-coil and measured the temperature inside the coil by 
a thermo-couple. The point of demagnetization which he 
obtained is rather higher than that obtained by the platinum- 
tube or current-in-the-wire methods, but not so high as that 
given by Hopkinson (loc. cit.) who used a copper wire for 
his temperature measurements. More recently a paper has 
appeared by Morris (Phil. Mag. vol. xliv. Sept. 1897), who 
employed the same method of heating as Curie, but measured 
the temperature with a platinum wire. It will be seen that 
the method of heating with a coil is less perfect than with 

Resistance and Inductance of a Wire to a Discharge. 433 

a platinum tube, and less simple than the current in the 

The results of the foregoing experiments were communicated 
to Section Gof the British Association at Toronto, and a brief 
abstract appears in the B. A. Report for 1897, but the curves 
were not reproduced. 

In conclusion, I should like to thank Professor Oallendar 
for kind suggestions and other assistance. 

Macdonald Physics Laboratory, 
December 20, 1898. 

XL. The Equivalent Resistance and Inductance of a Wire 
to an Oscillatory Discharge. By Edwin H. Bakton, 
D.Sc, F.R.S.E., Senior Lecturer in Physics, University 
College, Nottingham *. 

I.N an article in the Philosophical Magazine for May 
1886 f, Lord Rayleigh, whilst greatly extending Max- 
well's treatment of the self-induction of cylindrical con- 
ductors, confined the discussion of alternating currents to 
those which followed the harmonic law with constant ampli- 
tude. The object of the present note is to slightly modify 
the analysis so as to include also the decaying periodic 
currents obtained in discharging a condenser and the case of 
the damped trains of high-frequency waves generated by a 
Hertzian oscillator and now so often dealt with experi- 
mentally. In fact, it was while recently working with the 
latter that the necessity of attacking this problem occurred 
to me. 

Resume of previous Theories. — To make this paper in- 
telligible without repeated references to both Maxwell and 
Rayleigh, it may be well to explain again the notation used 
and sketch the line of argument followed. 

The conducting wire is supposed to be a straight cylinder 
of radius a, the return wire being at a considerable distance. 
The vector potential, H, the density of the current, w, and 
the " electromotive force at any point " may thus be con- 
sidered as functions of two variables only, viz., the time, t, 
and the distance, r, from the axis of the wire. The total 
current, C, through the section of the wire, and the total 
electromotive force, E, acting round the circuit, are the 
variables whose relation is to be found. It is assumed that 

H=S + T + V 2 + .... +T,r* . . . (1) 

* Communicated by the Physical Society : read January 27, 1899. 
f " On the Self-induction and Resistance of Straight Conductors." 

434 Dr. E. H. Barton on the Equivalent Resistance and 

where S, T , T l5 &c, are functions of the time. A relation 
between the T's is next established so that the subscripts are 
replaced by coefficients. The value of H at the surface of the 
wire is equated to AC, where A is a constant. This leads to 
Maxwell's equation (13) of art. 690. The magnetic permea- 
bility, //,, of the wire, which Maxwell had treated as unity, is 
now introduced by Lord Rayleigh, who thus obtains in place 
of Maxwell's (14) and (15) the following equations : — 

^=^* + i^-^ + -;;- + ii^^--a? + ---->--' (2) 

AC-S = J + ^ + g:?;g +.... + lit2 , |t<n ,^+....j > (o) 

where a, equal to //R, represents the conductivity (for steady 
currents) of unit length of the wire. 
By writing 

»(«) = ! + «+ pr^-+ — + 1 2 [ / tiW 2 + - • W 

equations (2) and (3) are then transformed as follows 
ctt ~~ dt 

d$ A dC* ,( d\ dT 

-ty^dtj-dt' • • • w 

C = -^|).f: .... (6) 
we have further 

5-^ (7) 

I- dt [n 

Lord Rayleigh then applies equations (5), (6), and (7) to 
sustained periodic currents following the harmonic law, where 
all the functions are proportional to e ipt , and obtains 

E=R'C+^L'C, (8) 

R / and 1/ denoting the effective resistance and inductance 
respectively to the currents in question. The values of R' 
and \J are expressed in the form of infinite series. For high 
frequencies, however, they are put also in a finite form, 
since, when p is very great, equation (4) reduces analytically 

*W=F7 = V' (9) 

£ V7T •*• 

* AC is printed here in Fhil. Mag., May 1886, p. 387 ; but appears to 

be a slip for A — . 
v dt 

Inductance of a Wire to an Oscillatory Discharge. 435 
so that 


= xh (10) 

Equivalent Resistance and Inductance for Oscillatory Dis- 
charges. — To effect the object of this paper we must now 
apply equations (5), (6), and (7) to the case of logarithmically- 
damped alternating currents where all the functions are 
proportional to e^~ k)pi . 

The value of E so obtained must then be separated into 
real and imaginary parts as in (8), and then, together with 
the imaginary quantities, must be collected a proportionate 
part of the real ones so as to exhibit the result in the form 

E = B"C+(e-%L // C (11) 

The quantities denoted by B/' and \J' in this equation will 

then represent what may be called the equivalent resistance 

and inductance of length / of the wire to the damped periodic 

currents under discussion. For, the operand being now 

e d-k)pt^ jfa Q time differentiator produces (i — k)p, and not ip 

simply as in equation (8) for the sustained harmonic currents. 

Thus (5), (6), and (7), on elimination of 8, Z, and -jj, give 

E _/;_ k \ n „\ | <Kip*p-kp*fi) (m 

Now we have 


^)- 1+ 2-l2 + 48-180 + *-" * {l6) 

cj)(ipufJL — /cpa/jb) 
<fi (ipufjL — kjjafA) 

+i-[ip^+ |^v- 1 -^p z «V - m ]^py w } . (u) 

Hence, substituting (14) in (12) and collecting the terms 
as in (ll), we find that 

R" 1 + i 2 s A(l + A«) 3 , 3 l-2F-3/t 4 4 4 

TT =1 + T2-AV+ — jj-W fgo AV • ••-, (15) 

= l+ToP 2a V-f«nAV...., . . (17) 

436 Dr. E. H. Barton on the Equivalent Resistance and 

or J- (16) 

L«=z[A+Mi+ l^-^^^V-^j^^V..-)] J 

Putting £ = in these equations and denoting by single- 
dashed letters the corresponding values of the resistance and 
inductance, we have 

-t*' 1 i ■*■ 2 2 2 

s =i+_^v- lTo 


L'=Z[A + Ki-^*V...-)], • • (18) 

which are Lord Rayleigh's well-known formulae * for periodic 
currents of constant amplitude. 

By taking the differences of the resistances and inductances 
with damping and without, we have at once 

*L^K =k yaY+ ^^-V«V+ (19) 


L''-L=^(j-.^+|V"V ••••)• • • (20) 

These show that if the frequency is such that a few terms 
sufficiently represent the value of the series, then both resist- 
ance and inductance are increased by the damping. 

High- Frequency Discharges. — Passing now to cases where 
p is very great, as in the wave-trains in or induced by a 
Hertzian primary oscillator, we have from equation (10), 

where s= y/l + k 2 and cot6 = k. 

On substituting this value of <p/4>' in equation (12) and 
collecting as before, we obtain the solution sought, viz.: 

^q = (*H*ps d )* cos 2+(i—k)pya A + *s/a/j,s/p cos A; (22) 

whence R" , ,. x 6 nON 

-g- = (a/ips z ^cos-^, ^23) 

Equations (19) and (20), p. 387, loc. cit. 

Inductance of a Wire to an Oscillatory Discharge. 437 

and L" . , , , y 

-r^ =aA+ (<zfLS/p)» COS^ 

MHSWf ' ' ' <21> 

Discussion of the Results for High Frequencies. 
On putting & = 0, in equations (23) and (24), to reduce to 


the case of sustained simple harmonic waves, 5=1, 0=--\ 


whence, denoting by single dashes these special values of R" 
and L", we obtain 

g- = Sfatp ; (25) 

"-'{*+</£} > • • • • <*> 

which are Lord Rayleigh's high-frequency formulae*. 

Referring again to equations (23) and (24), we see that 
for a given value of _p, if k varies from to co , the factor 
involving s increases without limit while that involving 
increases to unity. Hence, with increasing damping, it 
appears that R" and U f each increase also, while ever the 
equations remain applicable. Now an infinite value of k 
involves zero frequency f. And a certain large, though 
finite, value of k would prevent the frequency being classed as 
u high." 

Dividing equation (23) by (25) gives 

g'=(2 s 3 )icos| = K S a y (27) 

Thus, for a given value of k, the ratio W/Rf is independent 
of the frequency of the waves. It is therefore convenient to 
deal with K a function of k only, rather than with R"/R 
which is a function of p also. 

Differentiating to k, we have 

1Tr ok cos 77 + sin -r 
1k~ 711 ' (28) 

* Equations (26) and (27), p. 390, loc. cit. 

t This follows from the fact that electric currents or waves generated 
by an oscillatory discharge may be represented by e~ kpt cos pt, in which 
Jcp is finite, so k is infinite only when p is zero. 

Phil. Mag. S. 5. Vol. 47. No. 288. May 1899. 2 H 

438 Dr. E. H. Barton on the Equivalent Resistance and 

Fig. 1. — Exhibiting graphically K = R"/R' as a function of 
k, the damping factor. 

flUH ■■ 

Inductance of a Wire to an Oscillatory Discharge. 439 

which is positive for all values of k from to go , hence K 
increases continuously with k. For & = 0, this becomes 


. dk A =0 ~~ 




which assists in plotting K as a function of k. 
Differentiating again, we obtain 

dk 2 

, (7 + 3*«)cos-g + 2*sin 5 } 




Since this expression is positive for all values of k from 
to oo , we see that K plotted as a function of k is a curve 
which is always convex to the axis of k. Thus the nature of 
R'yR' as a function of k is sufficiently determined. 

Pairs of corresponding values of K and k for a few typical 
cases are shown in the accompanying Table, and part of the 
curve coordinating them is given in fig. 1. It is not necessary 
to plot much of the curve, as only a small part of it can 
apply to any actual case. For, although k may have any 
positive value up to go , the high values of k, as already men- 
tioned, correspond to low values of p and so exclude them 
from the application of the high-frequency formula. 

Table showing the values of K = R"/B/, the ratio of equiva- 
lent resistances to waves with damping and without. 

Damping Factor, 
h = cot 0. 

Subsidiary quantities involved. 

Ratio of 
K=^R fR . 


6 - 2 = l-fP. 




i- =0-0798 


42° 44' 


T044 nearly 

J* =00955 


42° 16' 


1-054 „ 

i =01595 

40° 28' 


1097 „ 

— =0-319 


36° 9' 


1-228 „ 


22° 30' 


2-197 „ 


13° 17' 


4 602 „ 


9° 13' 



2 H 2 

440 Equivalent Resistance of a Wire to Oscillatory Discharge* 

Fig. 2. — Instantaneous Form of Wave- train for fc = l, whence 
R"/R / = 2-197. 

Figure 2 shows the form of a wave-train for which k = l 
and K=2*197. That is to say, in this extreme case where 
all the functions vary as e^' 1 ^, and the wave-train passing 
a given point of the wire is accordingly represented by 
e-^cospt, then the equivalent resistance is 2*197 times that 
which would obtain for simple harmonic waves uniformly 
sustained and of the same frequency. 

Figure 3 represents the form of the wave-trains generated 
and used in some recent experiments on attenuation*. In 

this case the value of k was approximately vtc— , or the 

logarithmic decrement per wave =27rk = m 6, and the corre- 
sponding value of K, the ratio of Bf'/R', is T054. Now in 

* "Attenuation of Electric Waves along a Line of Negligible Leakage," 
Phil. Mag. Sept. 1898, pp. 296-805. 

Diffraction Fringes and Micrometric Observations. 441 

Fig. 3. — Instantaneous Form of Wave-Train for &='3/7r, whence 
R'VR' = ] -0.54. 


the experiments just referred to the frequency was '65 X 10° 
per second, and R'/R became 31*6. Hence R"/R has the 
value 31*6 x 1-054=33-3 nearly. Thus, writing e - w ^ 2Lv for 
the attenuator of the waves along the wires instead of 
4f g -EV/2L» " ncreases the index by about five and a half per cent., 
and so brings it by that amount nearer to the value deter- 
mined experimentally. 

Univ. Coll., Nottingham, 
Nov. 29, 1898. 

XLI. On certain .Diffraction Fringes as applied to Micro- 
metric Observations. By L. N. Gr. Filon, M.A., Demon- 
strator in Applied Mathematics and Fellow of University 
College, Londonf. 

1. rTVEE following paper is largely criticism and exten- 
X sion of Mr. A. A. Michelson's memoir "On the 
Application of Interference Methods to Astronomical Obser- 
vations," published in the Phil. Mag. vol. xxx. p. 256, March 

* See Equation (2) p. 301, Phil. Mag. Sept. 1898. 

t Communicated by the Physical Society : read November 2o, 1898. 

442 Mr. L. N. G. Filon on certain Diffraction Fringes 

Light from a distant source is allowed to pass through two 
thin parallel slits. The rays are then focussed on a screen 
(or the retina of the eye) and interference-fringes are seen. 
If the distant source be really double, or extended, the fringes 
will disappear for certain values of the distance between the 
slits. This distance depends on the angle subtended by the 
two components of the double source or the diameter of the 
extended source. 

Mr. Michelson, however, in obtaining his results treated 
the breadth of the slits as small compared with the wave- 
length of light and their length as infinite. This seems un- 
justifiable a priori. The present investigation takes the 
dimensions of the slits into account. 

2. Suppose we have an aperture or diaphragm of any 
shape in a screen placed just in front of the object-glass of a 
telescope (fig. 1). 

Fig. 1. 

s* — - 

, -/ , 

< *: 




/ J^' 



Let the axis of the telescope be the axis of z. Let the 
axes of oc and y be taken in the plane of the diaphragm OQ 
perpendicular to and in the plane of the paper respectively. 
Let S be a source of light whose coordinates are U, V, W. 
Let Q be any point in the diaphragm whose coordinates are 
(#, y). Let A P be a screen perpendicular to the axis of the 
telescope, and let (p, q) be the coordinates of any point P on 
this screen. Let A'P' be the conjugate image of the screen 
AP in the object-glass. 

Let b = distance of centre C of lens from screen AP. 

b'= „ „ „ „ image of screen AP. 

/= distance of diaphragm OQ from plane A'P'. 

Then if, as is usual, we break up a wave of light coming 

as applied to Micrometric Observations. 443 

from S at the diaphragm, the secondary wave due to the 
disturbance at Q would have to travel along a path QRTP in 
order to reach a point P on the screen, being regularly 

But since P f is the geometrical image of P, all rays which 
converge to P (i. e. pass through P) after refraction, must 
have passed through P' before refraction, to the order of our 

Hence the ray through Q which is to reach P must be 


Moreover, P and P' being conjugate images the change of 
phase of a wave travelling from P' to P is constant to the first 
approximation and independent of the position of Q. 

Now the disturbance at P due to an element dx dy of the 
diaphragm at Q is of the form 

^^ sin 2^ -SQ-QR-m. BT-TP), 

where X is the wave-length, t is the period, A is a constant, 
(i is the index of refraction of the material of the lens, and h 
is put instead of QP outside the trigonometrical term, because 
the distance of the lens from the diaphragm and the inclination 
of the rays are supposed small. 

But P'Q + QR + fiUT + TP = constant for P. 

Therefore the disturbance 


= A^ sin ^^ -SQ + P'Q -const.) 

SQ 2 =(.z-U) 2 + G/-V)*-r-W 3 

P # Q , =(*+^) f +(y+^) i +/ s - 

Now in practice x, y, p, q are small compared with b\ b, f, 
or W ; U and V are small compared with W. Neglecting 
terms of order IP/W 3 , xW/W, & c ., we find 

In like manner 

FQ-Z/a ft— 4- ^ + ? 2 4-^4- W 4- L f!±ff! 

remembering that / is very nearly equal to V because the 
diaphragm is very close to the lens. 

441: Mr. L. N. G. Filon on certain Diffraction Fringes 

Hence the difference of retardation measured by length in 

P'Q-SQ=const. +(| + ^)» + (f + \)y 

+*(*+/>(? -If)- 

If now the geometrical image of S lie on the screen AP 
{i.e. if the screen is in correct focus) b' = W and the last 
term disappears. 

If, however, the screen be out of focus \jV is not equal to 
1/W, and the term in x^+y 2 may be comparable with the 
two others, if b' be not very great compared with b. Thus 
we see that appearances out of focus will introduce expres- 
sions of the same kind as those which occur when no lenses 
are used. 

We will, however, only consider the case where the screen 
is in focus. Let — u and — v be the coordinates of the geome- 
trical image of S ; then 

u/b = TJ/W, v/b=Y/W. 

The difference of retardation measured by length in air is 
therefore of the form 

Hence the total disturbance at P (integrating over the two 
slits) is given by the expression 

a+k h 

a -k —h 
-a + k h 

C j Cj ' 27r / xt . P + n . 9 + v \ 


■a — k 

where A = a constant, 

2a = distance between centres of slits, 
2 k = breadth of either slit, 
21i — length of either slit. 

This, being integrated out, gives 

^ 2ALX . 2irt 27rq+v . 2irq+v, . Zirp + u, 

D=— 57 — i — \7 — r~\ sm — cos— -^-^— asm— ^— Asm r ~* h, 
ir 2 (p-\-u)(q + v) r X b X b X b 

as applied to Micrometric Observations. 445 

whence the intensity of light 

4A 2 i 2 A* 2™ , . . 9 2*k, N . ,2tA, 

I = ~T7 — , \2? — i — ^2 cos "FT W + w ) Sln T^ - (? + v ) sm ~TT" (P 

This may be written 

64AW 2™ , x Sm Tx ^ v) Bm !x ( ^ + " } 

This gives fringes parallel to x and y : k being very small 
compared with a, the quick variation term in v is 

2 27ra , . 
° TxT ^ + ^- 

Consider the other two factors, namely : 

sin^-7— (g -f r) sm 2 -^— (p + u) 


sin <27 
If we draw the curve y= — %~ (see fig. 2), we see that 


these factors are only sensible, and therefore their product is 
only sensible, for values of p + u and q + v which are numeri- 
cally less than bX/2h and bX/2k respectively. 

Fig. 2. 

Hence the intensity becomes very small outside a rectangle 
whose centre is the geometrical image and whose vertical 
and horizontal sides are bX/k and bX/h respectively. 

This rectangle I shall refer to as the " visible " rectangle of 
the source. 

Inside this rectangle are a number of fringes, the dark 
lines being given by 

q -f- v = — -. bX 

* 4« 

and the bright ones by q+ v = nbX/2a. 

446 Mr. L. N. G. Filon on certain Diffraction Fringes 

The successive maximum and minimum intensities do not 
vary with a. Hence, what Mr. Michelson calls the measure 
of visibility of the fringes, namely the quantity 

I, -Is 


where I l5 1 2 are successive maximum and minimum intensities, 
does not vary with the distance between the slits. The only 
effect of varying the latter is to make the fringes close up or 
open out. Hence for a point-source of light the fringes 
cannot be made to practically disappear. 

3. Consider now two point-sources of light whose geo- 
metrical images are J 1? J 2 , and draw their 
(fig. 3). 

FIr. 3. 

visible rectangles 

: J 


To get the resultant intensity we have to add the in- 
tensities at every point due to each source separately. 

Then it may be easily seen that the following are the 
phenomena observed in the three cases shown in fig. 3: — 

(1) The two sets of fringes distinct. Consequently no 
motion of the slits can destroy the fringes. In this case, 
however, the eye can at once distinguish between the two 
sources and Michelson's method is unnecessary. 

(2) Partial superposition : the greatest effect is round the 
point K, where the intensities due to the two sources are 
very nearly equal. If v' — v be the distance between Jj and 
J 2 measured perpendicularly to the slits, so that (v' — v)/b is 
the difference of altitude of the two stars when the slits are 
horizontal, then over the common area the fringe system is 
(a) intensified if v' —v be an even multiple of bX/Aa, (b) 
weakened, or even destroyed, if v' — v be an odd multiple of 
b\/4:a. For in case (a) the maxima of one system are super- 
posed upon the maxima of the other, while in case (b) the 
maxima of the one are superposed upon the minima of the 
other. This common area, however, will contain only com- 
paratively faint fringes, the more distinct ones round the 

as applied to Micro metric Observations. 447 

centres remaining unaffected. We may suppose case (2) to 
occur whenever the centre of either rectangle lies outside the 
other, i. e. whenever v , — v>b\/2k, u / — u>b\/2h,u' — u being 
the horizontal distance between J l and J 2 . 

(3) Almost complete superposition of the visible rectangles. 
The fringes of high intensity are now affected. These are 
destroyed or weakened whenever a is an odd multiple of 
b\/4:(v f — v), provided that the intensity of one source be not 
small compared with that of the other. 

Case (3) may be taken to occur when v' — v <b\/2k and 
u' ' — u< b\/2h. 

The smallest value of a for which the fringes disappear is 
b\/i(v f -v). 

If v r —v be very small, this may give a large value of a. 

Now a double star ceases to be resolved by a telescope of 
aperture 2r if (v' — v)/b<\/2r, and when this relation holds 
the smallest value of a for which the fringes disappear is not 
less than r/2, which is the greatest separation of the slits 
which can conveniently be used. Hence the method ceases 
to be available precisely at the moment when it is most 

(4) Mr. Michelson, in the paper quoted above, noticed this 
difficulty, and described an apparatus by means of which the 
effective aperture of the telescope could be indefinitely 
increased. He has not shown, however, that the expression 
for the disturbance remains of the same form, to the order of 
approximation taken, and he has made no attempt to work 
out the results when the slit is taken of finite width, as it 
should be. 

In his paper Mr. Michelson describes two kinds of appa- 
ratus. I shall confine my attention to the second one, as 
being somewhat more symmetrical. 

So far as I can gather from Mr. Michelson's description, 
the instrument consists primarily of a system of three mirrors 
a, b, c and two strips of glass e, d (fig. 4). The mirrors a and 
b are parallel, and c, d, e are parallel. Light from a point P 
in one slit is reflected at Q and R by the mirrors a and b, is 
refracted through the strip e, and finally emerges parallel to 
its original direction as T U. Light from a point P' in the 
other slit is refracted through the strip d, and reflected at S' 
and T' by the strips e and c. 

I may notice in passing that the strip e should be half 
silvered, but not at the back, for if the ray S' T is allowed to 
penetrate inside the strip and emerge after two refractions 
and one reflexion, not only is a change of phase introduced, 
owing to the path in the glass, which complicates the analysis, 

448 Mr. L. N. Gr. Filon on certain Diffraction Fringes 

but tlie conditions of reflexion, which should be the same for 
all four mirrors, are altered, and this changes the intensities 
of the two streams. We shall see afterwards that this silver- 
ing can be done without impeding the passage of the trans- 
mitted stream, as it will turn out thnt the two streams must 
be kept separate. 


Fig. 4. 
__n\q_. — W 


A %l — 'Vm¥\ 

!__— — - — — ~~" 


c i^ 

Suppose then that a plane wave of light whose front is 
M M / is incident upon the diaphragm. Let us break the wave 
up, as is usual, in the plane of the diaphragm. Let Z be a 
point on the screen whose cordinates are (p, q) at which the 
intensity of light is required. 

Then if C be the centre of the object-glass, the direction in 
which rays T U, T' U' must proceed in order to converge to 
Z after refraction is parallel to C Z. 

Hence PQ, RS, TU, P'Q', R'S', T'U' are ail parallel to OZ, 
and the direction-cosines of CZ are 

*V + ? 2 + 6 2 ' s/'pi + q^ + tf' ^zfi + tf + o* 

I shall assume that the strips e and d are cut from the same 
plate and are of equal thickness. This will sensibly simplify 
the analysis, though, as I think, it would not materially influ- 
ence the appearances if the strips were unequal. 

If, however, we suppose them equal, we may neglect the 
presence of strips, as far as refraction is concerned, since 
clearly the retardation introduced is the same for all parallel 

If now UU f be a plane perpendicular to CZ, then, since we 
know that rays parallel to TU, T'U' converge to a focus at Z, 
the only parts of the paths of the rays which can introduce a 

as applied to Micrometric Observations. -449 

difference of phase are 

MP + PQ + QR + RS + TU for one stream, 

M'P' + PQ' + R'S' + S'T' + T'U' for the other stream. 

Produce PQ, P'Q' to meet it in V and V and let N, N' be 
the feet of the perpendiculars from R and S' on PQ, T'U' 

Thus we may take the change of phase as due to the 

(MP + PV)+(NQ + QR) 
for diffraction at one slit, and to the retardation 
(M'P + P V) + (ST + T'N') 

for rays proceeding from the other slit. 

The terms in the first brackets give us the expression which 
we had before, viz. : — 

As to the other terms 

NQ + QR = QR (l + cos26)==«^i^ r ?^=2acosW), 

v T COS (p 

where a is the distance between the mirrors a, b and <£> is the 
angle of incidence of any ray on these mirrors. 

Similarly S'F + T'N' = 2/3 cos i/r, where /3 is the distance 
between e and c and ty the angle of incidence of any ray 
upon e and c. 

Now if the mirrors a and b are inclined to the plane of the 
diaphragm at an angle 0, c, d, e at an angle ( — &), then 

, q sin + b cos 
cos <p — 

cos yjr- 

— q sin & + b cos 0' 

\/p 2 + q 2 + tf 
To find the disturbance at Z we have 

a + k + h 

Ci Cy A . 27r/\t p + u , q + v _ , \ 

a — k —h 
-a+k +h 

, ( \i Cj A . 27r / Xt p + u q + v rt ~ ,\ 


450 Mr. L. N. G. Filon on certain Diffraction Fringes 

which, on being integrated, gives 

2kbX . 2-rrq + Vj . 2-rrp + u, . 2ir/Xt \ 2iry 

"27 — : \ f , — >sm -r- - T ™ A: sm — ^— h sin — ( e Icos , 

7r 2 (p-j-u)(q+v) X b \ b \\t J X 

where — €-y=— V -Jl a -2ftcosf, _ 6 + 7= ?; -±-£ a _2«cos<£, 

whence e = a cos <£ -f ft cos >/r, 

7= — t— a + p cos >/r — a cos </>. 

Hence the intensity I of light at (p, q) is 

ir (p + u)*(q + vr X b X b X 

, v + 2 Scos 6' — a cos z (ft sin 0' -\- a sin 0) 

where 7= -^-^a + ~ / o b— x a 

b x /f + q 2 + b 2 </ p 2 + q * + b * 9- 

In the last term we may put — y =^ = |-, for if we went 

Vi? 2 + g 2 + b 2 o 

to a higher approximation, we should introduce cubes of p/b, 
qlb which we have hitherto neglected. 

If, further, we make ft cos 6' = a cos 0, which can always he 
managed without difficulty, the second term, which would 
contain squares on expansion, disappears and we have 

(v+ q)a — ( ft sin 0' + asin 6)q 
7 _ 

= \va + q(a-(ft sin 0' + a sin 6))\/b. 

This gives fringes of breadth bX/2(a— (ft sin & + a sin 0)). 
These may be reckoned from the bright fringe 7 = ; i. e. 

— va 

qQ = a-{ftsm6' + *sm0)' 

The visibility of the fringes for a single source will, as before, 
not be affected by changing a : for a second source the origin 
of the fringes is given by 

q '=—v'a/(a—^ft sin 0' + asin 0^). 

and if the visible rectangles overlap, there will be a sensible 
diminution of the fringe appearance whenever 

q - q Q ' = (n + i)bX/2\a- (ft sin 0